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Question 21Question →

If xx and yy are positive integers, is x3−y3x^3 - y^3 divisible by 33?

(1) x+2yx + 2y is divisible by 33.
(2) x2−y2x^2 - y^2 is divisible by 33.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The question asks whether x3−y3x^3 - y^3 is divisible by 3. By Fermat's Little Theorem or factoring a3−a=a(a−1)(a+1)a^3 - a = a(a-1)(a+1) (the product of three consecutive integers, always divisible by 3), any integer cubed has the same remainder when divided by 3 as the integer itself (a3≡a(mod3)a^3 \equiv a \pmod 3). Therefore, x3−y3x^3 - y^3 is a multiple of 3 if and only if x−yx - y is a multiple of 3. Statement (1) specifies that x+2yx + 2y is divisible by 3. We can rewrite x+2yx + 2y as (x−y)+3y(x - y) + 3y. Because 3y3y is automatically a multiple of 3, (x−y)(x - y) must also be divisible by 3. This yields a definitive 'Yes', so Statement (1) alone is sufficient. Statement (2) states that x2−y2=(x−y)(x+y)x^2 - y^2 = (x - y)(x + y) is divisible by 3. Because 3 is prime, this requires 3 to divide either (x−y)(x - y) or (x+y)(x + y). If x=4,y=1x = 4, y = 1, 3 divides (x−y)=3(x - y) = 3, giving a 'Yes'. If x=2,y=1x = 2, y = 1, 3 divides (x+y)=3(x + y) = 3 while (x−y)=1(x - y) = 1, giving x3−y3=7x^3 - y^3 = 7, which is a 'No'. Hence, Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question stem using algebraic modular properties.
Since a3≡a(mod3)a^3 \equiv a \pmod 3 for any integer aa, x3−y3≡x−y(mod3)x^3 - y^3 \equiv x - y \pmod 3. Thus, x3−y3x^3 - y^3 is divisible by 3 if and only if x−yx - y is divisible by 3.
Simplifying the target expression reduces the problem to determining whether 3 divides (x−y)(x - y).
2
Evaluate Statement (1): x+2yx + 2y is divisible by 3.
x+2y=(x−y)+3yx + 2y = (x - y) + 3y. Since 3y3y is always a multiple of 3, (x+2y)(x + 2y) is divisible by 3 if and only if (x−y)(x - y) is divisible by 3.
This guarantees a definitive 'Yes' to the target question. Statement (1) alone is SUFFICIENT.
3
Evaluate Statement (2): x2−y2x^2 - y^2 is divisible by 3.
x2−y2=(x−y)(x+y)x^2 - y^2 = (x - y)(x + y). Since 3 is prime, 3 must divide (x−y)(x - y) or (x+y)(x + y). If x=4,y=1x = 4, y = 1, then x2−y2=15x^2 - y^2 = 15 (divisible by 3) and x3−y3=63x^3 - y^3 = 63 (divisible by 3) -> YES. If x=2,y=1x = 2, y = 1, then x2−y2=3x^2 - y^2 = 3 (divisible by 3), but x3−y3=7x^3 - y^3 = 7 (NOT divisible by 3) -> NO.
Statement (2) yields both 'Yes' and 'No' cases. Statement (2) alone is NOT SUFFICIENT.

Key Concept

Divisibility and Modular Congruences in Integer Properties
Estimated Time:2m 0s
Question 22Question →

The table below presents operational performance metrics for six properties within a boutique hotel portfolio:

Property NameRegionOccupancy Rate (%)Average Daily Rate ($)Customer Satisfaction (1–10)
Grand AzureCoastal823409.1
Alpine HavenMountain754108.8
Urban VantageMetropolitan882908.4
Solstice ResortCoastal684509.3
Metro EliteMetropolitan913108.6
Highland LodgeMountain793809.0

Which of the following statements regarding the portfolio data are true? Select all that apply.

Select all that apply

Show answer & explanation

Answer: Among properties with an Average Daily Rate greater than 350,exactly2haveaCustomerSatisfactionscoreofatleast9.0.;NopropertyintheMountainregionhasanOccupancyRateexceeding80350, exactly 2 have a Customer Satisfaction score of at least 9.0.; No property in the Mountain region has an Occupancy Rate exceeding 80% OR an Average Daily Rate below 350.; Exactly 4 properties satisfy at least one of the following criteria: Occupancy Rate greater than 85% OR Average Daily Rate greater than $400.

Answer

The correct statements are those identifying 2 properties with ADR over 350andsatisfactionofatleast9.0,statingthatnoMountainpropertyexceeds80350 and satisfaction of at least 9.0, stating that no Mountain property exceeds 80% occupancy or has ADR below 350, and noting that exactly 4 properties satisfy either an occupancy over 85% or an ADR over $400.
The correct options properly execute multi-column boolean criteria (AND vs. OR). Specifically: filtering properties with ADR over 350forsatisfactionatorabove9.0correctlyisolates2properties;evaluatingMountainregionpropertiesconfirmszerosatisfyeitheroccupancyover80350 for satisfaction at or above 9.0 correctly isolates 2 properties; evaluating Mountain region properties confirms zero satisfy either occupancy over 80% or ADR under 350; and combining properties with occupancy over 85% or ADR over $400 correctly identifies a subset of 4 properties.

Step-by-Step Solution

1
Evaluate the statement regarding Occupancy Rate ≥80%\geq 80\% AND Customer Satisfaction ≥8.5\geq 8.5.
Properties meeting Occupancy ≥80%\geq 80\% are Grand Azure (82%), Urban Vantage (88%), and Metro Elite (91%). Of these, Urban Vantage has a satisfaction of 8.4 (<8.5< 8.5). Only 2 properties meet both criteria, making this statement false.
Conjunctive filtering requires both conditions to hold simultaneously.
2
Evaluate the statement regarding properties with ADR >$350> \$350 having Customer Satisfaction ≥9.0\geq 9.0.
Properties with ADR >$350> \$350 are Alpine Haven ($410\$410), Solstice Resort ($450\$450), and Highland Lodge ($380\$380). Solstice Resort (9.3) and Highland Lodge (9.0) meet the satisfaction threshold (2 properties total), making this statement true.
First filter the subset by ADR threshold, then apply the satisfaction criterion.
3
Evaluate the statement regarding Mountain region properties having Occupancy >80%> 80\% OR ADR <$350< \$350.
Alpine Haven (75%, $410\$410) and Highland Lodge (79%, $380\$380) both fail both conditions. Therefore, 0 Mountain properties satisfy the OR condition, making the statement true.
Disjunctive filtering checks if at least one condition is met per property.
4
Evaluate the statement regarding Coastal properties having Occupancy ≥70%\geq 70\% AND ADR ≥$350\geq \$350.
Grand Azure has ADR $340<$350\$340 < \$350, and Solstice Resort has Occupancy 68%<70%68\% < 70\%. Since neither satisfies both criteria, the statement is false.
A universal condition requires all items in the subset to satisfy the full compound logic.
5
Evaluate the statement regarding properties with Occupancy >85%> 85\% OR ADR >$400> \$400.
Occupancy >85%> 85\%: Urban Vantage, Metro Elite. ADR >$400> \$400: Alpine Haven, Solstice Resort. Combining these gives 4 unique properties, making the statement true.
An inclusive OR filter pools all elements that satisfy either criterion.

Key Concept

Conditional Filtering and Selection
Question 23Question →

To decrease total operating expenses, the Fairview Municipal Water District plans to install bio-purification filters upstream of its primary membrane filtration system. Water district officials note that microplastic debris currently clogs and damages the delicate membrane filters, requiring $2 million annually in replacement and repair costs. Because the new bio-purification filters effectively capture over 90 percent of incoming microplastics before they reach the primary membranes, officials conclude that installing these bio-filters will significantly reduce the facility's overall annual operating costs.

Which of the following, if true, most seriously weakens the water district officials' argument?

Show answer & explanation

Answer: The chemical solutions required to routinely clean and maintain the bio-purification filters cost substantially more per year than the total annual expenditure for replacing damaged membrane filters.

Answer

The chemical solutions required to routinely clean and maintain the bio-purification filters cost substantially more per year than the total annual expenditure for replacing damaged membrane filters.
The conclusion asserts that adding bio-purification filters will reduce overall annual operating expenses by eliminating 2millioninmembranerepairs.Thislogicreliesontheassumptionthatrunningthebio−filterswillnotgeneratenewexpensesequaltoorgreaterthan2 million in membrane repairs. This logic relies on the assumption that running the bio-filters will not generate new expenses equal to or greater than 2 million. The correct response points out that routine maintenance chemicals for the bio-filters cost substantially more per year than the $2 million saved, demonstrating that the net financial outcome will be an increase in total operating expenses, directly weakening the conclusion.

Step-by-Step Solution

1
Deconstruct the argument structure
Premise: Microplastics cause $2 million in annual damage to membrane filters. Premise: Bio-filters remove 90% of microplastics before reaching membranes. Conclusion: Installing bio-filters will reduce overall annual operating costs.
Identifying the central claim and supporting evidence reveals the underlying assumption.
2
Identify the unstated assumption
The argument assumes that the ongoing cost to purchase, operate, and maintain the new bio-filters will be significantly less than the $2 million currently spent on membrane repairs.
A net reduction in total operating costs requires that savings exceed any new expenses incurred by the plan.
3
Evaluate choices for an undermining fact
The statement showing that chemical cleaning solutions for the bio-filters cost more per year than the $2 million membrane repair bill directly breaks the underlying assumption.
If new operational expenses exceed current repair costs, total operating expenses will increase rather than decrease.

Key Concept

Evaluating Plan Feasibility and Net Cost-Benefit Assumptions
Question 24Question →

To reduce regional water consumption, an agricultural cooperative plans to distribute automated soil-moisture sensors to all member farms. The sensors send real-time alerts whenever soil moisture drops below an optimal threshold, allowing farmers to irrigate only when necessary rather than on a fixed calendar schedule. The cooperative's directors conclude that installing these sensors will significantly decrease total agricultural water usage across the region over the next two years. Which of the following, if true, most seriously weakens the directors' argument?

Show answer & explanation

Answer: Most member farmers currently irrigate less than the optimal moisture threshold due to water cost concerns, but plan to increase watering to recommended sensor levels once alerts are automated.

Answer

The directors' argument is most seriously weakened by the finding that most member farmers currently underwater their crops relative to the optimal threshold and intend to increase watering once sensor alerts are implemented.
The conclusion relies on the unstated assumption that current schedule-based irrigation uses more water than sensor-guided optimal irrigation. The correct choice reveals that farmers currently apply less water than the optimal threshold due to costs, and will increase watering once sensor feedback is available. This counter-behavior causes total water consumption to rise, directly undermining the cooperative's objective.

Step-by-Step Solution

1
Identify the argument's premises and conclusion.
Premise: Automated sensors alert farmers to irrigate only when moisture drops below an optimal threshold, replacing fixed schedules. Conclusion: Regional agricultural water usage will decrease significantly.
Understanding the logical jump from premise to conclusion reveals the implicit assumption.
2
Uncover the central unstated assumption.
The argument assumes that farmers' current fixed schedules deliver more water than the sensor-determined optimal threshold requires.
If current watering is equal to or less than optimal levels, adopting precision sensors will not reduce water volume.
3
Evaluate the choices to find new evidence breaking this assumption.
The statement showing that farmers currently underwater their crops and plan to raise irrigation to match sensor recommendations directly refutes the assumption of net water savings.
If farmers increase watering frequency upon receiving sensor data, overall regional consumption will rise rather than fall.

Key Concept

Weakening Plan-to-Goal Arguments by Identifying Unintended Behavioral Counter-Effects
Question 25Question →

A wooden box contains 15 identical tokens, each marked with a distinct integer from 11 to 1515, inclusive. If one token is drawn at random from the box, what is the probability that the integer on the drawn token is a prime number?

Show answer & explanation

Answer: 25\frac{2}{5}

Answer

The probability that the integer on the drawn token is a prime number is 25\frac{2}{5}.
The total number of possible outcomes when choosing one token from 1515 tokens is 1515. The prime numbers between 11 and 1515, inclusive, are 2,3,5,7,11,2, 3, 5, 7, 11, and 1313. There are 66 favorable outcomes. The probability is therefore 615\frac{6}{15}, which simplifies to 25\frac{2}{5}.

Step-by-Step Solution

1
Determine the total number of possible outcomes (the denominator).
The total number of tokens is 1515, so N=15N = 15.
The sample space consists of all integers from 11 to 1515, inclusive.
2
Identify and count all prime numbers in the set from 11 to 1515, inclusive.
The prime numbers in this set are 2,3,5,7,11,2, 3, 5, 7, 11, and 1313. Thus, there are 66 prime numbers.
A prime number is an integer strictly greater than 11 that has exactly two distinct positive divisors: 11 and itself. Note that 11 is not prime.
3
Calculate the basic probability P(Prime)=Number of Favorable OutcomesTotal Number of Possible OutcomesP(\text{Prime}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}.
P(Prime)=615=25P(\text{Prime}) = \frac{6}{15} = \frac{2}{5}.
Dividing the favorable count 66 by total count 1515 and simplifying by dividing numerator and denominator by 33 yields 25\frac{2}{5}.

Key Concept

Basic Single-Event Probability and Definition of Prime Numbers
Estimated Time:45s
Question 26Question →

Set SS consists of all consecutive integers from −m-m to nn, inclusive, where mm and nn are positive integers with n>mn > m. If set SS contains exactly 2525 integers and the sum of all integers in set SS is 7575, what is the value of mm?

Show answer & explanation

Answer: 9

Answer

The value of mm is 99.
To find mm, use the two fundamental properties of consecutive integer sets: term count and set sum. The number of terms from −m-m to nn inclusive is n−(−m)+1=n+m+1=25n - (-m) + 1 = n + m + 1 = 25, which yields n+m=24n + m = 24. The sum of an evenly spaced set is the product of the number of terms and the arithmetic mean of the smallest and largest terms: 25×−m+n2=7525 \times \frac{-m + n}{2} = 75, which simplifies to n−m=6n - m = 6. Subtracting n−m=6n - m = 6 from n+m=24n + m = 24 gives 2m=182m = 18, so m=9m = 9.

Step-by-Step Solution

1
Set up the equation for the number of terms in the set.
n+m=24n + m = 24
The number of integers from −m-m to nn inclusive is n−(−m)+1=n+m+1=25n - (-m) + 1 = n + m + 1 = 25.
2
Set up the equation for the sum of the integers in the set.
n−m=6n - m = 6
The sum of an arithmetic progression is given by number of terms×mean=25×−m+n2=75\text{number of terms} \times \text{mean} = 25 \times \frac{-m + n}{2} = 75, leading to n−m2=3\frac{n - m}{2} = 3.
3
Solve for mm using the two linear equations.
m=9m = 9
Subtracting n−m=6n - m = 6 from n+m=24n + m = 24 yields 2m=182m = 18, giving m=9m = 9.

Key Concept

Properties of consecutive integer sets: inclusive term counting (n−start+1n - \text{start} + 1) and set sum calculation (count×mean\text{count} \times \text{mean}).
Estimated Time:1m 30s
Question 27Question →

A fuel tank initially contains 6060 gallons of a fuel mixture that is 10%10\% ethanol and 90%90\% gasoline by volume. A mechanic removes xx gallons of this mixture and replaces it with an equal volume of pure ethanol to obtain a new mixture that is 25%25\% ethanol by volume. What is the value of xx?

Show answer & explanation

Answer: 10

Answer

The volume of fuel mixture that must be removed and replaced with pure ethanol is 1010 gallons.
Replacing 1010 gallons of the 10%10\% ethanol fuel with pure ethanol removes 11 gallon of ethanol and adds 1010 gallons of pure ethanol. The total ethanol in the tank becomes 6−1+10=156 - 1 + 10 = 15 gallons, which represents exactly 25%25\% of the total 6060-gallon volume.

Step-by-Step Solution

1
Calculate the initial volume of ethanol in the tank
Ethanol volume = 0.10×60=60.10 \times 60 = 6 gallons.
Establishes the starting quantity of the solute.
2
Express the amount of ethanol after removal and replacement in terms of xx
Final ethanol volume = 6−0.10x+x=6+0.90x6 - 0.10x + x = 6 + 0.90x gallons.
Removing xx gallons of fuel removes 10%10\% ethanol, while adding xx gallons of pure ethanol adds 100%100\% ethanol.
3
Set up an equation using the target ethanol concentration
6+0.90x=0.25×60=156 + 0.90x = 0.25 \times 60 = 15.
The final 6060-gallon mixture must contain 25%25\% ethanol by volume.
4
Solve the equation for xx
0.90x=9  ⟹  x=100.90x = 9 \implies x = 10.
Isolates xx to find the required replacement volume.

Key Concept

Dilution and fluid replacement in mixture problems
Question 28Question →

If xx and yy are integers such that −5≤x≤−1-5 \le x \le -1 and 2≤y≤62 \le y \le 6, what is the minimum possible value of x−yx+y\frac{x - y}{x + y}?

Show answer & explanation

Answer: -11

Answer

The minimum possible value of the expression is -11.
The correct answer is -11 because the numerator x−yx - y is negative for all allowed values of xx and yy. To minimize a negative fraction, the denominator x+yx + y must be positive and minimized (equal to 1), while the numerator must be as negative as possible. Choosing x=−5x = -5 and y=6y = 6 gives a numerator of −11-11 and a denominator of 11, producing the minimum value of -11.

Step-by-Step Solution

1
Evaluate the sign of the numerator
Since xx is negative (x≤−1x \le -1) and yy is positive (y≥2y \ge 2), x−yx - y is always negative (x−y≤−3x - y \le -3).
Subtracting a positive integer from a negative integer yields a negative result.
2
Determine the conditions for minimizing a negative fraction
To obtain the minimum (most negative) value, the denominator x+yx + y must be a positive integer and as small as possible.
A negative numerator divided by a positive denominator yields a negative quotient. Dividing by a smaller positive number yields a quotient with larger absolute value, making it smaller (more negative).
3
Identify the smallest positive denominator and solve for variables
The smallest positive integer for x+yx + y is 11. Setting y=1−xy = 1 - x gives x−y=2x−1x - y = 2x - 1.
Since xx and yy are integers, x+yx + y must be an integer.
4
Maximize the magnitude of the negative numerator
Using the lowest boundary x=−5x = -5 gives y=6y = 6. The value of the expression is −5−6−5+6=−11\frac{-5 - 6}{-5 + 6} = -11.
Evaluating at x=−5x = -5 and y=6y = 6 gives numerator −11-11 and denominator 11, resulting in −11-11.

Key Concept

Minimizing algebraic fractions involving signed numbers
Question 29Question →

Read the argument below regarding conservation paleontology and synthetic genomics, and match each statement from the text to its corresponding logical role within the overall argument:

"While bioethicists argue that de-extinction efforts targeting apex predators could destabilize contemporary food webs, ecologists emphasize that reintroducing keystone herbivores such as the woolly mammoth would enhance arctic tundra biodiversity by compacting snow cover and suppressing soil warming. However, because funds allocated to synthetic genomics are diverted directly from active habitat protection programs for endangered living species, pursuing de-extinction ultimately undermines global biodiversity preservation goals."

Click a left item, then click its matching right item

Items

Bioethicists argue that de-extinction efforts targeting apex predators could destabilize contemporary food webs.
Reintroducing keystone herbivores such as the woolly mammoth would enhance arctic tundra biodiversity.
Funds allocated to synthetic genomics are diverted directly from active habitat protection programs for endangered living species.
Pursuing de-extinction ultimately undermines global biodiversity preservation goals.

Matches

Show answer & explanation

Answer

The statement regarding bioethicists matches the opposing ethical concern background; the statement about keystone herbivores matches the concession of potential ecological benefit; the statement regarding funding diversion matches the supporting empirical premise; and the statement that pursuing de-extinction ultimately undermines global preservation goals matches the author's main conclusion.
The argument uses a contrastive structure. It begins by introducing competing views (bioethicists' concerns vs. ecologists' cited benefits of tundra mammoth reintroduction) as context and concession. The author then introduces their own central position using 'However', concluding that pursuing de-extinction ultimately undermines global preservation goals because of resource diversion from active endangered species programs.

Step-by-Step Solution

1
Identify structural transition markers and the author's primary thesis
The pivot word 'However' shifts focus from potential benefits to the author's core position that pursuing de-extinction ultimately undermines preservation goals.
Main conclusions in Critical Reasoning frequently follow structural pivot words like 'however' or 'therefore' and synthesize the author's final verdict.
2
Analyze supporting premises directly feeding the main conclusion
The clause noting that funds for synthetic genomics are diverted from active habitat protection provides the explicit evidence for why global preservation is undermined.
Premises explain the 'why' behind the main claim.
3
Differentiate background context and concessions from the main claim
The opening bioethicist point serves as background context, while the ecologist point about mammoth tundra effects functions as a counter-perspective/concession acknowledged in the argument.
Distinguishing secondary claims and concessions prevents misidentifying intermediate or counter-points as the author's final stance.

Key Concept

Identifying Main Conclusions and Final Claims
Question 30Question →

A financial research study across 60 regional credit unions examined the implementation of machine-learning credit risk algorithms and subsequent small-business loan default rates. The study observed that credit unions implementing machine-learning algorithms experienced a 40 percent lower default rate on small-business loans over a three-year period than credit unions relying strictly on traditional manual underwriting. Skeptical financial analysts contend that the algorithm itself did not cause the lower default rate. Instead, they argue that credit unions adopting the algorithm had systematically raised their minimum credit score requirements for all loan applicants immediately prior to software installation, thereby selecting an inherently lower-risk applicant pool.

Based on the information provided, select the statement that most strengthens the financial analysts' alternative explanation, and select the statement that most weakens the financial analysts' alternative explanation (thereby supporting a direct causal relationship between algorithm adoption and reduced defaults).

Click a left item, then click its matching right item

Items

Strengthens Analysts' Alternative Explanation
Weakens Analysts' Alternative Explanation

Matches

Show answer & explanation

Answer

The statement strengthening the analysts' argument is that credit unions raising credit score requirements without the algorithm achieved equal default reductions. The statement weakening the analysts' argument is that default reductions occurred even among borrowers with credit scores below the new threshold.
To strengthen the analysts' claim that credit score threshold increases (and not algorithm installation) caused the reduction in defaults, we look for evidence showing that raising thresholds alone produces the same result. The statement noting that unions raising score thresholds without algorithms achieved equal default reductions directly confirms this alternative cause. To weaken the analysts' claim, we look for evidence where default reductions occurred independently of the higher credit score threshold. The statement showing default reductions among waiver-program borrowers below the threshold proves the algorithm worked even when the proposed confounding factor was absent.

Step-by-Step Solution

1
Analyze the causal claim and the counter-argument
Study Claim: Algorithm adoption causes lower default rates. Analysts' Claim: Tightened minimum credit score requirements (confounding variable), not the algorithm, caused lower default rates.
To strengthen or weaken an alternative causal explanation, we must isolate the proposed confounding variable (credit score threshold) from the primary variable (algorithm adoption).
2
Evaluate candidate statements to strengthen the analysts' argument
The statement regarding credit unions that raised credit score requirements without installing the algorithm shows that credit score tightening alone yields an identical default reduction. This proves the confounder is sufficient to explain the outcome without the algorithm.
A control group demonstrating that the outcome occurs to the same degree without the treatment directly strengthens the alternative explanation.
3
Evaluate candidate statements to weaken the analysts' argument
The statement showing default reductions among borrowers below the new credit score threshold demonstrates that the algorithm reduced defaults in a population where the credit score change was absent.
Demonstrating the effect in a subgroup unaffected by the proposed confounding variable directly undermines the claim that the confounder is responsible for the overall effect.

Key Concept

Evaluating Confounding Variables and Alternative Causal Explanations
Estimated Time:2m 30s
Question 31Question →

Let m=2a⋅53⋅11bm = 2^a \cdot 5^3 \cdot 11^b and n=24⋅5c⋅112n = 2^4 \cdot 5^c \cdot 11^2, where aa, bb, and cc are positive integers. If the greatest common divisor of mm and nn is 23⋅52⋅1122^3 \cdot 5^2 \cdot 11^2, and the least common multiple of mm and nn has exactly 180180 positive integer divisors, what is the value of a+b+ca + b + c?

Show answer & explanation

Answer: 13

Answer

The value of a+b+ca + b + c is 1313.
To find a,b,ca, b, c, analyze prime factor exponents. The GCD uses minimum exponents: min⁡(a,4)=3  ⟹  a=3\min(a,4)=3 \implies a=3, min⁡(3,c)=2  ⟹  c=2\min(3,c)=2 \implies c=2, and min⁡(b,2)=2  ⟹  b≥2\min(b,2)=2 \implies b \ge 2. The LCM uses maximum exponents: 24,53,11b2^4, 5^3, 11^b. The number of divisors of the LCM is (4+1)(3+1)(b+1)=20(b+1)=180(4+1)(3+1)(b+1) = 20(b+1) = 180, yielding b=8b=8. Adding a+b+c=3+8+2=13a+b+c = 3+8+2 = 13.

Step-by-Step Solution

1
Use the prime exponent rule for greatest common divisor to find aa and cc, and bound bb.
a=3a = 3, c=2c = 2, and b≥2b \ge 2.
The greatest common divisor takes the minimum exponent for each prime factor: min⁡(a,4)=3  ⟹  a=3\min(a, 4) = 3 \implies a = 3, min⁡(3,c)=2  ⟹  c=2\min(3, c) = 2 \implies c = 2, and min⁡(b,2)=2  ⟹  b≥2\min(b, 2) = 2 \implies b \ge 2.
2
Determine the prime factorization of lcm(m,n)\text{lcm}(m, n).
lcm(m,n)=24⋅53⋅11b\text{lcm}(m, n) = 2^4 \cdot 5^3 \cdot 11^b.
The least common multiple takes the maximum exponent for each prime factor: max⁡(3,4)=4\max(3, 4) = 4, max⁡(3,2)=3\max(3, 2) = 3, and max⁡(b,2)=b\max(b, 2) = b since b≥2b \ge 2.
3
Apply the total number of divisors formula to solve for bb.
b=8b = 8.
The number of positive integer divisors of 24⋅53⋅11b2^4 \cdot 5^3 \cdot 11^b is (4+1)(3+1)(b+1)=20(b+1)=180(4+1)(3+1)(b+1) = 20(b+1) = 180, which yields b+1=9b+1 = 9 so b=8b = 8.
4
Calculate the sum a+b+ca + b + c.
a+b+c=13a + b + c = 13.
Summing the values a=3a=3, b=8b=8, and c=2c=2 gives 3+8+2=133 + 8 + 2 = 13.

Key Concept

GCD and LCM prime factor exponent rules and total divisor formula
Question 32Question →

What is the remainder when the expression 745⋅322+8157^{45} \cdot 3^{22} + 8^{15} is divided by 1010?

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Answer: 5

Answer

The remainder is 5.
Dividing any number by 10 leaves a remainder equal to the units digit of that number. By analyzing the units digit pattern (cyclicity of period 4) for powers of 7, 3, and 8: 745≡71=7(mod10)7^{45} \equiv 7^1 = 7 \pmod{10}, 322≡32=9(mod10)3^{22} \equiv 3^2 = 9 \pmod{10}, and 815≡83=2(mod10)8^{15} \equiv 8^3 = 2 \pmod{10}. The expression simplifies to (7×9)+2=63+2=65(7 \times 9) + 2 = 63 + 2 = 65, which has a units digit of 5. Therefore, the remainder when divided by 10 is 5.

Step-by-Step Solution

1
Relate remainder modulo 10 to units digit cyclicity.
Finding the remainder when an expression is divided by 10 is equivalent to finding its units digit.
Any positive integer NN can be expressed as 10k+r10k + r, where rr is the units digit and the remainder when NN is divided by 10.
2
Determine the units digit of 7457^{45}.
745≡7(mod10)7^{45} \equiv 7 \pmod{10}.
Powers of 7 repeat their units digits in a cycle of length 4 (7, 9, 3, 1). Dividing the exponent 45 by 4 gives a remainder of 1, so 7457^{45} has the same units digit as 71=77^1 = 7.
3
Determine the units digit of 3223^{22}.
322≡9(mod10)3^{22} \equiv 9 \pmod{10}.
Powers of 3 repeat their units digits in a cycle of length 4 (3, 9, 7, 1). Dividing the exponent 22 by 4 gives a remainder of 2, so 3223^{22} has the same units digit as 32=93^2 = 9.
4
Determine the units digit of the product 745⋅3227^{45} \cdot 3^{22}.
745⋅322≡3(mod10)7^{45} \cdot 3^{22} \equiv 3 \pmod{10}.
The product of the units digits is 7×9=637 \times 9 = 63, which has a units digit of 3.
5
Determine the units digit of 8158^{15}.
815≡2(mod10)8^{15} \equiv 2 \pmod{10}.
Powers of 8 repeat their units digits in a cycle of length 4 (8, 4, 2, 6). Dividing the exponent 15 by 4 gives a remainder of 3, so 8158^{15} has the same units digit as 83=5128^3 = 512, which ends in 2.
6
Combine the results to find the final remainder modulo 10.
5
Adding the units digit of the first term (3) and the second term (2) gives 3+2=53 + 2 = 5.

Key Concept

Units Digit Cyclicity and Modular Arithmetic
Question 33Question →

If rr and ss are the two distinct real roots of the quadratic equation x2−6x+4=0x^2 - 6x + 4 = 0, what is the value of r3+s3r+s\frac{r^3 + s^3}{r + s}?

Show answer & explanation

Answer: 2424

Answer

The value of r3+s3r+s\frac{r^3 + s^3}{r + s} is 2424.
Using Vieta's formulas for x2−6x+4=0x^2 - 6x + 4 = 0, the sum of the roots is r+s=6r + s = 6 and the product is rs=4rs = 4. Factoring r3+s3r^3 + s^3 gives (r+s)(r2−rs+s2)(r + s)(r^2 - rs + s^2). Dividing by (r+s)(r + s) leaves r2−rs+s2r^2 - rs + s^2, which can be rewritten as (r+s)2−3rs(r + s)^2 - 3rs. Substituting the Vieta values yields 62−3(4)=36−12=246^2 - 3(4) = 36 - 12 = 24.

Step-by-Step Solution

1
Apply Vieta's formulas to the given quadratic equation x2−6x+4=0x^2 - 6x + 4 = 0.
The sum of the roots is r+s=6r + s = 6, and the product of the roots is rs=4rs = 4.
For a quadratic equation x2+bx+c=0x^2 + bx + c = 0, Vieta's formulas state that the sum of roots is −b-b and the product of roots is cc.
2
Factor the sum of cubes expression r3+s3r^3 + s^3.
r3+s3r+s=(r+s)(r2−rs+s2)r+s=r2−rs+s2\frac{r^3 + s^3}{r + s} = \frac{(r + s)(r^2 - rs + s^2)}{r + s} = r^2 - rs + s^2.
The sum of cubes factors algebraically into (r+s)(r2−rs+s2)(r + s)(r^2 - rs + s^2), and r+s=6≠0r + s = 6 \neq 0 allows cancellation.
3
Express r2−rs+s2r^2 - rs + s^2 in terms of (r+s)(r + s) and rsrs.
r2−rs+s2=(r+s)2−3rsr^2 - rs + s^2 = (r + s)^2 - 3rs.
Since (r+s)2=r2+2rs+s2(r + s)^2 = r^2 + 2rs + s^2, subtracting 3rs3rs yields r2−rs+s2r^2 - rs + s^2.
4
Substitute the known values r+s=6r + s = 6 and rs=4rs = 4 into the expression.
62−3(4)=36−12=246^2 - 3(4) = 36 - 12 = 24.
Evaluating the algebraic expression yields the final value.

Key Concept

Polynomial Factoring and Vieta's Formulas for Quadratic Equations
Estimated Time:1m 30s
Question 34Question →

If 5x−5x−2=60055^{x} - 5^{x-2} = 600\sqrt{5}, what is the value of xx?

Show answer & explanation

Answer: 4.5

Answer

The value of xx is 4.5.
Factoring out 5x−25^{x-2} converts the left side into 5x−2(25−1)=24⋅5x−25^{x-2}(25 - 1) = 24 \cdot 5^{x-2}. Dividing 6005600\sqrt{5} by 24 gives 25525\sqrt{5}, which equals 52.55^{2.5}. Setting x−2=2.5x - 2 = 2.5 gives x=4.5x = 4.5.

Step-by-Step Solution

1
Factor out the smallest power of 5, which is 5x−25^{x-2}, from the left side of the equation.
5x−2(52−1)=60055^{x-2}(5^2 - 1) = 600\sqrt{5}
Factoring isolates the constant multiplier from the variable power term.
2
Calculate the numerical value inside the parentheses.
24⋅5x−2=600524 \cdot 5^{x-2} = 600\sqrt{5}
52−1=25−1=245^2 - 1 = 25 - 1 = 24.
3
Divide both sides of the equation by 24.
5x−2=2555^{x-2} = 25\sqrt{5}
Isolating 5x−25^{x-2} gives 600524=255\frac{600\sqrt{5}}{24} = 25\sqrt{5}.
4
Rewrite 25525\sqrt{5} as a single exponential expression with base 5.
5x−2=52⋅50.5=52.55^{x-2} = 5^2 \cdot 5^{0.5} = 5^{2.5}
Using the product rule for exponents, 52⋅51/2=52+0.5=52.55^2 \cdot 5^{1/2} = 5^{2 + 0.5} = 5^{2.5}.
5
Equate the exponents since the bases are identical.
x−2=2.5  ⟹  x=4.5x - 2 = 2.5 \implies x = 4.5
For any positive base b≠1b \neq 1, if bm=bnb^m = b^n, then m=nm = n.

Key Concept

Exponents, Roots, and Powers of Integers
Question 35Question →

Let n=2a⋅3b⋅5n = 2^a \cdot 3^b \cdot 5, where aa and bb are positive integers. If nn has a total of 24 positive divisors and has an equal number of even positive divisors and odd positive divisors, what is the value of bb?

Show answer & explanation

Answer: 5

Answer

5
For n=2a⋅3b⋅51n = 2^a \cdot 3^b \cdot 5^1, the total number of divisors is (a+1)(b+1)(2)=24(a + 1)(b + 1)(2) = 24, giving (a+1)(b+1)=12(a + 1)(b + 1) = 12. The odd divisors are formed strictly from 3b⋅513^b \cdot 5^1, giving (b+1)(2)(b + 1)(2) odd divisors. The even divisors require at least one factor of 2, giving a(b+1)(2)a(b + 1)(2) even divisors. Setting even and odd divisor counts equal gives 2a(b+1)=2(b+1)2a(b + 1) = 2(b + 1), which reduces to a=1a = 1. Substituting a=1a = 1 into (a+1)(b+1)=12(a + 1)(b + 1) = 12 gives 2(b+1)=122(b + 1) = 12, leading directly to b=5b = 5.

Step-by-Step Solution

1
Set up the formula for total positive divisors of nn.
(a+1)(b+1)(1+1)=24  ⟹  (a+1)(b+1)=12(a + 1)(b + 1)(1 + 1) = 24 \implies (a + 1)(b + 1) = 12
The total number of divisors of p1e1p2e2⋯pkekp_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} is (e1+1)(e2+1)⋯(ek+1)(e_1 + 1)(e_2 + 1) \cdots (e_k + 1).
2
Determine the counts of odd and even positive divisors.
Odd divisors = 2(b+1)2(b + 1), Even divisors = 2a(b+1)2a(b + 1)
Odd divisors cannot contain any factors of 2 (exponent of 2 is 0). Even divisors must contain at least one factor of 2 (exponent of 2 can be 1,2,…,a1, 2, \dots, a).
3
Equate the number of odd and even divisors to find aa.
2a(b+1)=2(b+1)  ⟹  a=12a(b + 1) = 2(b + 1) \implies a = 1
Dividing both sides by 2(b+1)2(b + 1) (which is positive since b≥1b \ge 1) leaves a=1a = 1.
4
Solve for bb using the total divisor relation.
(1+1)(b+1)=12  ⟹  2(b+1)=12  ⟹  b=5(1 + 1)(b + 1) = 12 \implies 2(b + 1) = 12 \implies b = 5
Substituting a=1a = 1 into (a+1)(b+1)=12(a + 1)(b + 1) = 12 yields b=5b = 5.

Key Concept

Counting total, odd, and even positive divisors using prime factorization exponents
Question 36Question →

Determine the sum of all real solutions to the equation ∣x2−4x∣=3x−6|x^2 - 4x| = 3x - 6.

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Answer: 99

Answer

The sum of all valid real solutions is 99.
The correct answer is 99. Setting up the two cases x2−4x=3x−6x^2 - 4x = 3x - 6 and x2−4x=−(3x−6)x^2 - 4x = -(3x - 6) yields candidate roots x=1,6,3,x = 1, 6, 3, and −2-2. Because the absolute value expression ∣x2−4x∣|x^2 - 4x| cannot be negative, 3x−63x - 6 must be non-negative, requiring x≥2x \ge 2. Evaluating each candidate shows that x=1x = 1 and x=−2x = -2 produce negative right-hand sides and are extraneous. The only valid solutions are x=3x = 3 and x=6x = 6, whose sum is 3+6=93 + 6 = 9.

Step-by-Step Solution

1
Establish the domain condition for the right-hand side of the absolute value equation.
Since absolute values are non-negative, ∣x2−4x∣≥0|x^2 - 4x| \geq 0 requires 3x−6≥0  ⟹  x≥23x - 6 \geq 0 \implies x \geq 2.
An absolute value expression cannot equal a negative number.
2
Solve Case 1 where x2−4x=3x−6x^2 - 4x = 3x - 6.
x2−7x+6=0  ⟹  (x−1)(x−6)=0  ⟹  x=1x^2 - 7x + 6 = 0 \implies (x - 1)(x - 6) = 0 \implies x = 1 or x=6x = 6.
This corresponds to the positive branch of the absolute value.
3
Solve Case 2 where x2−4x=−(3x−6)x^2 - 4x = -(3x - 6).
x2−4x=−3x+6  ⟹  x2−x−6=0  ⟹  (x−3)(x+2)=0  ⟹  x=3x^2 - 4x = -3x + 6 \implies x^2 - x - 6 = 0 \implies (x - 3)(x + 2) = 0 \implies x = 3 or x=−2x = -2.
This corresponds to the negative branch of the absolute value.
4
Test all candidate solutions (x=−2,1,3,6x = -2, 1, 3, 6) against the domain constraint x≥2x \geq 2.
x=−2x = -2 yields 3(−2)−6=−12<03(-2)-6 = -12 < 0 (extraneous). x=1x = 1 yields 3(1)−6=−3<03(1)-6 = -3 < 0 (extraneous). x=3x = 3 yields ∣9−12∣=3=3(3)−6|9-12| = 3 = 3(3)-6 (valid). x=6x = 6 yields ∣36−24∣=12=3(6)−6|36-24| = 12 = 3(6)-6 (valid).
Extraneous roots introduced by unconstrained case splitting must be eliminated.
5
Sum the valid real solutions.
3+6=93 + 6 = 9.
The question asks specifically for the sum of all valid real solutions.

Key Concept

Absolute Value Equations and Extraneous Solution Verification
Question 37Question →

If nn is a positive integer such that 6n+3⋅152n−110n+1⋅9n+1=8100\frac{6^{n+3} \cdot 15^{2n-1}}{10^{n+1} \cdot 9^{n+1}} = 8{}100, what is the value of nn?

Show answer & explanation

Answer: 4

Answer

4
Rewriting all composite bases (6,15,10,96, 15, 10, 9) into prime bases (2,3,52, 3, 5) simplifies the equation to 4⋅3n⋅5n−2=8,1004 \cdot 3^n \cdot 5^{n-2} = 8,100. Dividing by 4 yields 3n⋅5n−2=2,025=34⋅523^n \cdot 5^{n-2} = 2,025 = 3^4 \cdot 5^2, which gives n=4n = 4.

Step-by-Step Solution

1
Express each composite base in the fraction using its prime factor decomposition.
Numerator: (2⋅3)n+3⋅(3⋅5)2n−1=2n+3⋅3n+3⋅32n−1⋅52n−1=2n+3⋅33n+2⋅52n−1(2 \cdot 3)^{n+3} \cdot (3 \cdot 5)^{2n-1} = 2^{n+3} \cdot 3^{n+3} \cdot 3^{2n-1} \cdot 5^{2n-1} = 2^{n+3} \cdot 3^{3n+2} \cdot 5^{2n-1}.
Denominator: (2⋅5)n+1⋅(32)n+1=2n+1⋅5n+1⋅32n+2(2 \cdot 5)^{n+1} \cdot (3^2)^{n+1} = 2^{n+1} \cdot 5^{n+1} \cdot 3^{2n+2}.
Breaking composite numbers down into prime factors allows terms with identical bases to be combined using exponent rules.
2
Simplify the fraction by subtracting the denominator exponents from the numerator exponents for each prime base.
2n+3⋅33n+2⋅52n−12n+1⋅32n+2⋅5n+1=2(n+3)−(n+1)⋅3(3n+2)−(2n+2)⋅5(2n−1)−(n+1)=22⋅3n⋅5n−2=4⋅3n⋅5n−2\frac{2^{n+3} \cdot 3^{3n+2} \cdot 5^{2n-1}}{2^{n+1} \cdot 3^{2n+2} \cdot 5^{n+1}} = 2^{(n+3)-(n+1)} \cdot 3^{(3n+2)-(2n+2)} \cdot 5^{(2n-1)-(n+1)} = 2^2 \cdot 3^n \cdot 5^{n-2} = 4 \cdot 3^n \cdot 5^{n-2}.
Applying the quotient rule for exponents: axay=ax−y\frac{a^x}{a^y} = a^{x-y}.
3
Set the simplified expression equal to 8,1008,100 and solve for nn.
4⋅3n⋅5n−2=8,100  ⟹  3n⋅5n−2=2,0254 \cdot 3^n \cdot 5^{n-2} = 8,100 \implies 3^n \cdot 5^{n-2} = 2,025. Prime factorization of 2,025=81⋅25=34⋅522,025 = 81 \cdot 25 = 3^4 \cdot 5^2. Equating exponents gives n=4n = 4 (and n−2=2n-2 = 2, which confirms n=4n=4).
Because 3 and 5 are distinct prime bases, the unique prime factorization theorem guarantees that 3n⋅5n−2=34⋅523^n \cdot 5^{n-2} = 3^4 \cdot 5^2 implies n=4n = 4.

Key Concept

Simplifying exponential expressions using prime factorization and quotient laws
Estimated Time:2m 0s
Question 38Question →

For any positive integer nn, let f(n)f(n) denote the product of all distinct prime factors of nn. For example, f(12)=2×3=6f(12) = 2 \times 3 = 6. What is the value of f(26⋅34⋅52−24⋅32⋅54)f(2^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4)?

Show answer & explanation

Answer: 330

Answer

330
To find f(26⋅34⋅52−24⋅32⋅54)f(2^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4), first factor out the greatest common factor 24⋅32⋅522^4 \cdot 3^2 \cdot 5^2. This yields 24⋅32⋅52(22⋅32−52)=24⋅32⋅52(36−25)=24⋅32⋅52⋅112^4 \cdot 3^2 \cdot 5^2 (2^2 \cdot 3^2 - 5^2) = 2^4 \cdot 3^2 \cdot 5^2 (36 - 25) = 2^4 \cdot 3^2 \cdot 5^2 \cdot 11. The distinct prime factors present in this expression are 2, 3, 5, and 11. Multiplying these distinct prime factors gives 2×3×5×11=3302 \times 3 \times 5 \times 11 = 330.

Step-by-Step Solution

1
Factor out the greatest common term 24⋅32⋅522^4 \cdot 3^2 \cdot 5^2 from 26⋅34⋅52−24⋅32⋅542^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4
24⋅32⋅52⋅(22⋅32−52)2^4 \cdot 3^2 \cdot 5^2 \cdot (2^2 \cdot 3^2 - 5^2)
Factoring out common prime powers simplifies the expression and avoids large calculations.
2
Evaluate the arithmetic expression inside the parentheses
22⋅32−52=4⋅9−25=36−25=112^2 \cdot 3^2 - 5^2 = 4 \cdot 9 - 25 = 36 - 25 = 11
11 is itself a prime number.
3
Write the full prime factorization of the overall number
24⋅32⋅52⋅1112^4 \cdot 3^2 \cdot 5^2 \cdot 11^1
All bases (2, 3, 5, 11) are prime numbers, giving the complete prime factorization.
4
Multiply each distinct prime factor together to evaluate f(n)f(n)
2×3×5×11=3302 \times 3 \times 5 \times 11 = 330
The function f(n)f(n) takes the product of each unique prime factor exactly once.

Key Concept

Prime Factorization and Product of Distinct Prime Factors
Question 39Question →

A set SS consists of kk consecutive integers. The sum of the first mm integers in set SS is 3434, and the sum of the last mm integers in set SS is 7474. If the sum of all kk integers in set SS is 189189, what is the value of kk?

Show answer & explanation

Answer: 14

Answer

The total number of integers in set S is 14.
By applying the property that the arithmetic mean of an evenly spaced set is the average of the mean of its first mm elements and the mean of its last mm elements, we establish that the mean of the set is 54m\frac{54}{m}. Since the total sum is 189189, k⋅54m=189k \cdot \frac{54}{m} = 189, which yields k=3.5mk = 3.5m. Substituting this ratio into the difference between the two subset sums m(k−m)=40m(k-m) = 40 gives 2.5m2=402.5m^2 = 40, so m=4m = 4 and k=14k = 14.

Step-by-Step Solution

1
Set up algebraic expressions for the sums of the first m terms and last m terms.
Let the set be S={a,a+1,…,a+k−1}S = \{a, a+1, \dots, a+k-1\}. The first mm terms sum to S1=ma+m(m−1)2=34S_1 = m a + \frac{m(m-1)}{2} = 34. The last mm terms sum to S2=m(a+k−m)+m(m−1)2=74S_2 = m(a+k-m) + \frac{m(m-1)}{2} = 74.
Consecutive integer sums can be represented by the starting term and the number of terms.
2
Subtract the sum of the first m terms from the sum of the last m terms.
S2−S1=m(a+k−m)−ma=m(k−m)=74−34=40S_2 - S_1 = m(a+k-m) - ma = m(k-m) = 74 - 34 = 40.
Subtracting eliminates the initial term aa and quadratic term m(m−1)2\frac{m(m-1)}{2}, giving a clean relationship between mm and kk.
3
Determine the arithmetic mean of set S using subset averages.
The average of the first mm terms is 34m\frac{34}{m} and the average of the last mm terms is 74m\frac{74}{m}. The average of the entire set is the midpoint of these two averages: Mean=12(34m+74m)=54m\text{Mean} = \frac{1}{2}\left(\frac{34}{m} + \frac{74}{m}\right) = \frac{54}{m}.
In any evenly spaced set, the overall median/mean is equal to the average of the lower-bound subset mean and upper-bound subset mean.
4
Relate the total sum to the set size k and overall mean.
Stotal=k×Mean  ⟹  189=k(54m)  ⟹  km=18954=3.5  ⟹  k=3.5mS_{total} = k \times \text{Mean} \implies 189 = k \left(\frac{54}{m}\right) \implies \frac{k}{m} = \frac{189}{54} = 3.5 \implies k = 3.5m.
The sum of a set of consecutive integers is always equal to the number of terms times the mean of the set.
5
Solve for m and k using the system of equations.
Substitute k=3.5mk = 3.5m into m(k−m)=40  ⟹  m(2.5m)=40  ⟹  2.5m2=40  ⟹  m2=16  ⟹  m=4m(k-m) = 40 \implies m(2.5m) = 40 \implies 2.5m^2 = 40 \implies m^2 = 16 \implies m = 4. Thus, k=3.5×4=14k = 3.5 \times 4 = 14.
Since m>0m > 0, taking the positive square root gives m=4m = 4, which leads directly to k=14k = 14.

Key Concept

Average and Sum Equivalences in Evenly Spaced Sets
Question 40Question →

How many integer values of yy satisfy the inequality ∣2y−7∣≤∣y+2∣|2y - 7| \le |y + 2|?

Show answer & explanation

Answer: 8

Answer

The total number of integer values of yy satisfying the inequality is 8.
Squaring both sides of ∣2y−7∣≤∣y+2∣|2y - 7| \le |y + 2| gives 3y2−32y+45≤03y^2 - 32y + 45 \le 0, which factors into (3y−5)(y−9)≤0(3y - 5)(y - 9) \le 0. The solution range for yy is 53≤y≤9\frac{5}{3} \le y \le 9. Since 53≈1.67\frac{5}{3} \approx 1.67, the integer values of yy satisfying this range are 2,3,4,5,6,7,8,92, 3, 4, 5, 6, 7, 8, 9, giving a total of 8 integers.

Step-by-Step Solution

1
Square both sides of the inequality ∣2y−7∣≤∣y+2∣|2y - 7| \le |y + 2|
(2y−7)2≤(y+2)2(2y - 7)^2 \le (y + 2)^2
Since both sides of an absolute value expression are non-negative, squaring both sides maintains the inequality direction.
2
Expand terms and move all terms to the left side
3y2−32y+45≤03y^2 - 32y + 45 \le 0
Expanding gives 4y2−28y+49≤y2+4y+44y^2 - 28y + 49 \le y^2 + 4y + 4. Subtracting (y2+4y+4)(y^2 + 4y + 4) from both sides produces the standard quadratic inequality.
3
Factor the quadratic expression to find critical points
(3y−5)(y−9)≤0(3y - 5)(y - 9) \le 0, yielding 53≤y≤9\frac{5}{3} \le y \le 9
The roots are y=53y = \frac{5}{3} and y=9y = 9. A quadratic with a positive leading coefficient is non-positive between its roots.
4
Determine all integers within the range [53,9]\left[\frac{5}{3}, 9\right]
2,3,4,5,6,7,8,92, 3, 4, 5, 6, 7, 8, 9 (8 integers total)
Because 53≈1.67\frac{5}{3} \approx 1.67, the smallest integer within the range is 2 and the largest is 9.

Key Concept

Solving absolute value inequalities of the form ∣A∣≤∣B∣|A| \le |B| by squaring both sides and determining integer solutions.
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