Tüm alıştırma soruları

Soru 621Soru →

In an effort to lower operating costs, the municipal transit authority of Oakhaven plans to replace its entire fleet of diesel-powered buses with electric buses. Management projects that this transition will significantly decrease total annual maintenance costs for the transit system, because electric buses have fewer moving mechanical parts than diesel buses do.

Which of the following is an assumption on which the municipal transit authority's projection depends?

The electricity required to charge the new electric buses will be generated entirely from renewable energy sources.

The transit authority will not experience any passenger fare revenue losses during the period when the bus fleet is being replaced.

The ongoing maintenance costs for the electronic and battery components of electric buses will not exceed the savings achieved from having fewer mechanical parts.

Diesel bus manufacturers will increase the cost of replacement parts for diesel buses over the next several years.

The municipal transit authority currently spends more on fuel each year than it spends on bus fleet maintenance.

Cevabı ve açıklamayı göster

Cevap: The ongoing maintenance costs for the electronic and battery components of electric buses will not exceed the savings achieved from having fewer mechanical parts.

Cevap

The ongoing maintenance costs for the electronic and battery components of electric buses will not exceed the savings achieved from having fewer mechanical parts.

The argument concludes that total annual maintenance costs will decrease because electric buses have fewer moving mechanical parts. For this to hold true, any new or existing maintenance costs associated with non-mechanical systems (such as batteries, software, and electrical wiring) must not outweigh the savings realized from reduced mechanical maintenance. If non-mechanical maintenance expenses exceed those mechanical savings, the overall maintenance budget will not decrease.

Adım Adım Çözüm

1

Deconstruct the argument into premise and conclusion.

Premise: Electric buses have fewer moving mechanical parts than diesel buses. Conclusion: Replacing the fleet with electric buses will significantly decrease total annual maintenance costs.

Isolating the premise and conclusion reveals the scope gap between mechanical parts and total maintenance costs.

2

Identify the underlying logical gap.

The author assumes that savings gained from having fewer mechanical parts will not be offset or surpassed by new non-mechanical maintenance expenses specific to electric buses.

Total maintenance costs encompass all components of a vehicle, not just mechanical ones.

3

Apply the Negation Test to verify the assumption.

Logical Negation: Non-mechanical maintenance costs WILL exceed the savings achieved from having fewer mechanical parts. Outcome: If this negated statement is true, total maintenance costs will increase or stay the same, completely destroying the conclusion.

A valid assumption must be necessary for the conclusion to hold true.

Anahtar Kavram

Identifying Unstated Assumptions

Soru 622Soru →

Under the 2025 Financial Technology Modernization Act, any proprietary trading firm operating automated execution algorithms in Market Region Z must maintain an immutable audit trail of all order routing decisions. During the third quarter of 2025, every trading firm operating in Market Region Z that experienced an algorithmic order routing failure was subjected to an immediate full-scale regulatory compliance audit. Furthermore, during that same quarter, no firm subjected to a full-scale regulatory compliance audit had maintained an immutable audit trail of all order routing decisions.

If the statements above are true, which of the following must also be true on the basis of them?

During the third quarter of 2025, every trading firm in Market Region Z that experienced an algorithmic order routing failure failed to maintain an immutable audit trail of all order routing decisions.

During the third quarter of 2025, any trading firm in Market Region Z that was subjected to a full-scale regulatory compliance audit must have experienced an algorithmic order routing failure.

Maintaining an immutable audit trail of order routing decisions is sufficient to prevent algorithmic order routing failures from occurring in Market Region Z.

Most trading firms operating in Market Region Z during the third quarter of 2025 failed to comply with the Financial Technology Modernization Act.

No trading firm in Market Region Z that maintained an immutable audit trail during the third quarter of 2025 experienced any regulatory scrutiny.

Cevabı ve açıklamayı göster

Cevap: During the third quarter of 2025, every trading firm in Market Region Z that experienced an algorithmic order routing failure failed to maintain an immutable audit trail of all order routing decisions.

Cevap

During the third quarter of 2025, every trading firm in Market Region Z that experienced an algorithmic order routing failure failed to maintain an immutable audit trail of all order routing decisions.

The correct answer is a direct, necessary deduction from the passage's two main facts. The stimulus establishes that every firm in Market Region Z experiencing a routing failure in Q3 2025 was audited, and that no audited firm in that quarter had maintained an immutable audit trail. Linking these two facts creates a solid chain of logic: any firm experiencing a routing failure in Q3 2025 necessarily fell into the group of audited firms, none of which had maintained an immutable audit trail.

Adım Adım Çözüm

1

Identify and break down the explicit factual premises provided in the stimulus.

Premise 1: Routing Failure (Q3 2025) → Subjected to Full-Scale Regulatory Compliance Audit. Premise 2: Full-Scale Regulatory Compliance Audit (Q3 2025) → Did NOT Maintain Immutable Audit Trail.

Establishing formal conditional relationships allows for precise logical synthesis without relying on unstated assumptions.

2

Synthesize the conditional chain using the transitive property.

Routing Failure (Q3 2025) → Full-Scale Audit → Did NOT Maintain Immutable Audit Trail. Therefore, Routing Failure (Q3 2025) → Did NOT Maintain Immutable Audit Trail.

Combining Premise 1 and Premise 2 yields a necessary conclusion that must be true based solely on the provided facts.

3

Evaluate the answer choices to identify the option that strictly matches this deduction while eliminating assumptions and speculations.

The statement asserting that every firm with a routing failure failed to maintain an immutable audit trail strictly expresses this valid deduction.

Valid GMAT inferences must follow necessarily from the premises with 100% certainty.

Anahtar Kavram

Distinguishing Inferences from Assumptions and Speculations

Tahmini Süre:2m 0s

Soru 623Soru →

A municipal water utility recently deployed high-precision acoustic sensors across its underground pipe network to identify micro-leaks early. Over the following year, repair crews successfully located and sealed 45 percent more micro-leaks than in any previous year. Paradoxically, despite these prompt repairs, the total volume of treated water lost throughout the municipal distribution network increased by 15 percent over the same period. Which of the following, if true, most helps to resolve the apparent paradox described above?

Operating the acoustic monitoring system required raising hydraulic pressure in the main distribution trunklines, which drastically increased the rate of water escaping from undetected major structural ruptures elsewhere in the system.

The utility spent significantly more money on specialized equipment calibration and sensor maintenance than it saved through reduced water treatment costs.

Repair crews required specialized training to interpret the data generated by the acoustic sensors, which initially slowed down the repair time for individual micro-leaks.

Neighboring municipal water utilities that did not install acoustic micro-leak sensors also reported substantial increases in total water losses over the same period.

Micro-leaks account for a smaller proportion of total pipeline damage in older urban water systems than in newly constructed suburban systems.

Cevabı ve açıklamayı göster

Cevap: Operating the acoustic monitoring system required raising hydraulic pressure in the main distribution trunklines, which drastically increased the rate of water escaping from undetected major structural ruptures elsewhere in the system.

Cevap

The statement explaining that operating the acoustic monitoring system required raising hydraulic pressure in the main distribution trunklines, thereby increasing the flow rate of undetected major structural ruptures, resolves the paradox.

The correct answer reconciles both facts by introducing a operational consequence of the technology. Raising system hydraulic pressure caused undetected high-volume leaks to spill significantly more water, overshadowing the water saved by fixing micro-leaks.

Adım Adım Çözüm

1

Identify the two apparently contradictory facts presented in the passage.

Fact 1: 45 percent more micro-leaks were located and sealed. Fact 2: Total volume of water lost in the network increased by 15 percent.

Resolving a paradox requires finding a piece of new information that allows both facts to be true simultaneously without denying either premise.

2

Evaluate the causal interaction introduced by each answer choice.

Increasing hydraulic pressure across the network to run the sensors creates a side effect: major undetected leaks lose water at a much faster rate, outweighing the savings from repaired micro-leaks.

This accounts for both the successful repair of micro-leaks and the overall increase in total volume lost.

Anahtar Kavram

Resolving Paradoxes through Confounding Side Effects and Volume Discrepancies

Soru 624Soru →

For any integer $n$ , the expression $n^2 + n + 7$ is an odd integer.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is true because

n(n + 1)

is the product of two consecutive integers, which is always even. Adding the odd constant

7

to an even integer yields an odd integer for all integer values of

n

.

The statement is true because

n^2 + n = n(n + 1)

represents the product of two consecutive integers and is therefore always even. Adding

7

(an odd integer) to an even integer always yields an odd integer regardless of whether

n

is positive, negative, or zero.

Adım Adım Çözüm

1

Factor the algebraic terms

n^2 + n

.

n^2 + n + 7 = n(n + 1) + 7

Factoring out

n

reveals the product of two consecutive integers.

2

Determine the parity of the product of two consecutive integers

n(n + 1)

.

n(n + 1)

is always even.

Of any two consecutive integers, exactly one is even. The product of an even integer and any integer is always even.

3

Determine the overall parity of the expression by adding

7

.

\text{even} + 7 = \text{odd}

Adding an odd integer to an even integer results in an odd integer.

Anahtar Kavram

The product of two consecutive integers is always even. Adding an odd integer to an even integer results in an odd integer.

Tahmini Süre:45s

Soru 625Soru →

What is the units digit of $2^{35}$ ?

2

4

6

8

0

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The units digit of

2^{35}

is 8.

The units digit of powers of 2 repeats every 4 powers in the sequence 2, 4, 8, 6. Dividing the exponent 35 by 4 yields a quotient of 8 and a remainder of 3. A remainder of 3 corresponds to the third number in the repeating sequence, which is 8.

Adım Adım Çözüm

1

Find the cyclicity pattern of the units digits for powers of 2.

The units digits of

2^1, 2^2, 2^3, 2^4, 2^5, \dots

are

2, 4, 8, 6, 2, \dots

, repeating in a cycle of length 4.

Units digits of powers follow a repeating periodic pattern determined by base arithmetic modulo 10.

2

Divide the exponent by the length of the cycle.

35 \div 4 = 8

with a remainder of

3

.

The remainder indicates how far into the 4-term repeating cycle the exponent 35 reaches.

3

Match the remainder to the corresponding term in the units digit cycle.

A remainder of 3 corresponds to the 3rd term in the cycle

(2, 4, 8, 6)

, which is 8.

The 3rd power in the repeating pattern gives a units digit of

2^3 = 8

.

Anahtar Kavram

Units Digit Cyclicity

Tahmini Süre:1m 0s

Soru 626Soru →

If $a$ , $b$ , and $c$ are integers such that $a^2 + 3b$ is an odd integer, $b^2 + 5c$ is an even integer, and $c^2 + 7a$ is an odd integer, which of the following expressions MUST be an even integer?

a(b + c)

a + b + c

a + bc

ab + c

ac + b

Cevabı ve açıklamayı göster

Cevap: $a(b + c)$

Cevap

The expression

a(b + c)

MUST be an even integer.

The expression

a(b + c)

is guaranteed to be an even integer because in both valid parity configurations—whether

a

is odd with

b, c

even, or

a

is even with

b, c

odd—the sum

(b + c)

is always even, making the product

a(b + c)

even.

Adım Adım Çözüm

1

Analyze the parity relationships from the given expressions.

Since

a^2

has the same parity as

a

,

3b

has the same parity as

b

,

b^2

has the same parity as

b

,

5c

has the same parity as

c

,

c^2

has the same parity as

c

, and

7a

has the same parity as

a

, the given conditions simplify to: (1)

a + b

is odd, (2)

b + c

is even, and (3)

c + a

is odd.

Multiplying an integer by an odd constant or raising an integer to a positive integer power does not change its parity.

2

Determine the valid parity configurations for

a

,

b

, and

c

.

Condition (1) implies

a

and

b

have opposite parities. Condition (2) implies

b

and

c

have the same parity. Condition (3) implies

c

and

a

have opposite parities. This yields two valid cases: Case 1:

b

and

c

are even, while

a

is odd. Case 2:

b

and

c

are odd, while

a

is even.

A sum of two integers is odd when they have opposite parities, and even when they have the same parity.

3

Evaluate the target expression

a(b + c)

across both valid cases.

In Case 1 (

a

odd,

b

even,

c

even):

b + c

is even, so

a(b + c) = \text{odd} \times \text{even} = \text{even}

. In Case 2 (

a

even,

b

odd,

c

odd):

b + c

is even, so

a(b + c) = \text{even} \times \text{even} = \text{even}

.

Any integer multiplied by an even integer yields an even integer.

Anahtar Kavram

Systematic parity deduction across multiple constraints and testing expressions under all valid parity configurations.

Soru 627Soru →

What is the smallest positive integer greater than $1$ that leaves a remainder of $1$ when divided by each of $4$ , $6$ , and $8$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The correct answer is 25.

The least common multiple of 4, 6, and 8 is 24, which is the smallest positive integer divisible by all three numbers. To leave a remainder of 1 upon division by each, we add 1 to 24, resulting in 25.

Adım Adım Çözüm

1

Calculate the least common multiple (LCM) of 4, 6, and 8.

\text{LCM}(4, 6, 8) = 24

Any positive integer divisible by 4, 6, and 8 must be a multiple of their least common multiple.

2

Add the desired remainder of 1 to the LCM.

24 + 1 = 25

Adding 1 to a common multiple guarantees that division by 4, 6, or 8 yields a remainder of 1.

Anahtar Kavram

Least Common Multiple (LCM) and Remainder Properties

Soru 628Soru →

Two positive integers $a$ and $b$ are in the ratio $3 : 8$ . If the least common multiple (LCM) of $a$ and $b$ is $360$ , what is the greatest common divisor (GCD) of $a$ and $b$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The greatest common divisor of

a

and

b

is 15.

Any two positive integers in the ratio

3 : 8

can be written as

3g

and

8g

, where

g

is their greatest common divisor. Because

3

and

8

are coprime (their GCD is

1

), the least common multiple of

3g

and

8g

is

3 \times 8 \times g = 24g

. Setting

24g = 360

and dividing by

24

yields

g = 15

.

Adım Adım Çözüm

1

Represent the two integers using their ratio and their greatest common divisor.

Let

g = \gcd(a, b)

. Then

a = 3g

and

b = 8g

, where

3

and

8

share no common factors other than

1

.

When two numbers are in reduced ratio

p : q

, dividing both by their GCD leaves coprime factors

p

and

q

.

2

Express the LCM of

a

and

b

in terms of

g

.

\text{LCM}(a, b) = 3 \times 8 \times g = 24g.

The LCM of two numbers

p \cdot g

and

q \cdot g

with

\gcd(p, q) = 1

is

p \cdot q \cdot g

.

3

Solve for

g

using the given LCM value of

360

.

24g = 360 \implies g = 15.

Dividing the given LCM by the product of the coprime ratio components yields the GCD.

Anahtar Kavram

Relationship between GCD, LCM, and coprime factor ratios of two positive integers

Tahmini Süre:1m 15s

Soru 629Soru →

A positive integer $n$ has only two distinct prime factors, $2$ and $3$ . If $n$ is a multiple of $12$ , is not divisible by $8$ , and has exactly $12$ positive divisors, what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 108

Cevap

The value of

n

is 108.

Because

n

has only

2

and

3

as prime factors, its prime factorization is

n = 2^a \times 3^b

. Divisibility by

12 = 2^2 \times 3^1

requires

a \ge 2

, and non-divisibility by

8 = 2^3

requires

a < 3

, which forces

a = 2

. Using the formula for total positive divisors,

(2+1)(b+1) = 12

, which simplifies to

3(b+1) = 12

and gives

b = 3

. Calculating

n = 2^2 \times 3^3

yields

108

.

Adım Adım Çözüm

1

Set up the prime factorization of

n

n = 2^a \times 3^b

where

a \ge 1

and

b \ge 1

The problem states that

2

and

3

are the only distinct prime factors of

n

.

2

Determine the exact value of exponent

a

a = 2

n

is divisible by

12 = 2^2 \times 3^1

(so

a \ge 2

) but not by

8 = 2^3

(so

a < 3

).

3

Determine the exact value of exponent

b

b = 3

The total number of positive divisors is

(a+1)(b+1) = (2+1)(b+1) = 3(b+1) = 12

, which solves to

b = 3

.

4

Calculate the value of

n

n = 108

n = 2^2 \times 3^3 = 4 \times 27 = 108

.

Anahtar Kavram

Determining integer values using prime factorization and the number of divisors formula

Soru 630Soru →

What is the sum of the distinct prime factors of $60$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The sum of the distinct prime factors of 60 is 10.

The prime factorization of

60

is

2^2 \times 3 \times 5

. The unique prime factors are

2

,

3

, and

5

. Adding these values together gives

2 + 3 + 5 = 10

.

Adım Adım Çözüm

1

Find the prime factorization of 60

60 = 2^2 \times 3^1 \times 5^1

Decompose 60 into prime factor powers.

2

List the distinct prime factors

The distinct prime factors are 2, 3, and 5

Focus only on the prime bases, ignoring exponents.

3

Sum the distinct prime factors

2 + 3 + 5 = 10

Add the unique prime numbers found in the factorization.

Anahtar Kavram

Prime Factorization and Distinct Prime Factors

Soru 631Soru →

Two positive integers $x$ and $y$ have a greatest common divisor (GCD) of $14$ and a least common multiple (LCM) of $840$ . If $14 < x < y < 200$ and $x$ is not a multiple of $4$ , what is the value of $y - x$ ?

42

98

112

126

238

Cevabı ve açıklamayı göster

Cevap: $98$

Cevap

The value of

y - x

is

98

.

The prime factorizations of

\text{GCD}(x,y)=14=2^1 \times 7^1

and

\text{LCM}(x,y)=840=2^3 \times 3^1 \times 5^1 \times 7^1

require

x

and

y

to split the exponents of

2, 3, 5, 7

. Since

x

is not a multiple of

4

,

x

contains

2^1

and

y

contains

2^3

. Both contain

7^1

. To satisfy

14 < x < y < 200

,

x

must take factor

5

(

x = 70

) and

y

must take factor

3

(

y = 168

). The difference is

168 - 70 = 98

.

Adım Adım Çözüm

1

Express the GCD and LCM in prime factorized form.

\text{GCD}(x, y) = 14 = 2^1 \times 3^0 \times 5^0 \times 7^1

and

\text{LCM}(x, y) = 840 = 2^3 \times 3^1 \times 5^1 \times 7^1

.

Prime factorization allows us to analyze the min and max prime exponents shared between

x

and

y

.

2

Determine the distribution of prime factor exponents between

x

and

y

.

For factor

7

: both

x

and

y

must have exponent

1

.
For factor

2

:

\min(a_x, a_y) = 1

and

\max(a_x, a_y) = 3

. Since

x

is not divisible by

4

(

2^2

),

a_x = 1

, which forces

a_y = 3

.
For factors

3

and

5

: one integer receives

3^1

and the other receives

5^1

.

The GCD takes the minimum exponent and the LCM takes the maximum exponent for each prime factor.

3

Test possible assignments of factors

3

and

5

under the given inequalities.

Case 1:

x

gets

3

and

y

gets

5

.

x = 2 \times 7 \times 3 = 42

,

y = 8 \times 7 \times 5 = 280

. This fails

y < 200

.
Case 2:

x

gets

5

and

y

gets

3

.

x = 2 \times 7 \times 5 = 70

,

y = 8 \times 7 \times 3 = 168

. This satisfies

14 < 70 < 168 < 200

.

Only Case 2 satisfies all structural and boundary constraints (

14 < x < y < 200

and

x

not a multiple of

4

).

4

Compute

y - x

.

y - x = 168 - 70 = 98

.

Direct subtraction of the valid integer values.

Anahtar Kavram

Prime Factorization Rules for GCD and LCM

Soru 632Soru →

Let $n$ be a positive integer whose only prime factors are $2$ , $3$ , and $5$ . If $\dfrac{n}{2}$ has $36$ positive divisors, $\dfrac{n}{3}$ has $36$ positive divisors, and $\dfrac{n}{5}$ has $32$ positive divisors, how many positive divisors does $n^2$ have?

96

180

216

245

343

Cevabı ve açıklamayı göster

Cevap: 245

Cevap

245

Writing

n

as

2^a \cdot 3^b \cdot 5^c

, the number of divisors of

\frac{n}{2}

,

\frac{n}{3}

, and

\frac{n}{5}

leads to the system

a(b+1)(c+1) = 36

,

(a+1)b(c+1) = 36

, and

(a+1)(b+1)c = 32

. Equating the first two yields

a = b

. Substituting

b = a

into the third gives

(a+1)^2 c = 32

. Since

a

is a positive integer,

(a+1)^2

must be a square dividing

32

, so

a+1 = 4 \implies a = 3

, giving

b = 3

and

c = 2

. Therefore,

n^2 = 2^6 \cdot 3^6 \cdot 5^4

, and the total number of positive divisors of

n^2

is

(6+1)(6+1)(4+1) = 245

.

Adım Adım Çözüm

1

Express

n

in terms of its prime factorization

n = 2^a \cdot 3^b \cdot 5^c

where

a, b, c \ge 1

The problem states that

2

,

3

, and

5

are the only prime factors of

n

.

2

Set up equations for the number of divisors of

\dfrac{n}{2}

,

\dfrac{n}{3}

, and

\dfrac{n}{5}

a(b+1)(c+1) = 36

,

(a+1)b(c+1) = 36

, and

(a+1)(b+1)c = 32

Dividing

n

by

2

,

3

, or

5

decreases the corresponding prime exponent by

1

. The formula for total positive divisors of

2^p 3^q 5^r

is

(p+1)(q+1)(r+1)

.

3

Solve the system of equations for exponents

a

,

b

, and

c

a = 3

,

b = 3

,

c = 2

Since

a(b+1)(c+1) = (a+1)b(c+1) = 36

, we get

a(b+1) = b(a+1) \implies a = b

. Substituting

b = a

into

(a+1)^2 c = 32

gives

(a+1)^2

as a factor of

32

. Testing perfect squares yields

a+1 = 4 \implies a = 3

, so

c = 2

. Checking in the first equation gives

3(4)(3) = 36

, which is consistent.

4

Determine the prime factorization and number of positive divisors of

n^2

n^2 = 2^6 \cdot 3^6 \cdot 5^4

, total divisors

= (6+1)(6+1)(4+1) = 7 \cdot 7 \cdot 5 = 245

Squaring

n

doubles each exponent in its prime factorization.

Anahtar Kavram

Prime Factorization and Number of Divisors Formula

Tahmini Süre:2m 0s

Soru 633Soru →

For a positive integer $N$ , the greatest common divisor of $N$ and $360$ is $120$ , and the least common multiple of $N$ and $450$ is $9000$ . What is the value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 3000

Cevap

The value of

N

is

3000

.

Prime factorizing

360

,

120

,

450

, and

9000

converts the GCD and LCM requirements into a system of min/max equations for the exponents of

2

,

3

, and

5

. The condition

\gcd(N, 360) = 120

dictates that the exponent of

3

in

N

must be exactly

1

, while the exponent of

2

is at least

3

. The condition

\text{lcm}(N, 450) = 9000

dictates that the exponent of

2

must be exactly

3

and the exponent of

5

must be exactly

3

. Combining these constraints gives

N = 2^3 \times 3^1 \times 5^3 = 3000

.

Adım Adım Çözüm

1

Find the prime factorizations of all known numbers

360 = 2^3 \times 3^2 \times 5^1

,

120 = 2^3 \times 3^1 \times 5^1

,

450 = 2^1 \times 3^2 \times 5^2

, and

9000 = 2^3 \times 3^2 \times 5^3

.

Prime factorization allows us to analyze GCD and LCM conditions using exponent minimums and maximums.

2

Apply the GCD exponent rule

\min(\text{exp}_N(p), \text{exp}_A(p)) = \text{exp}_{GCD}(p)

For factor

2

:

\min(a, 3) = 3 \implies a \ge 3

; For factor

3

:

\min(b, 2) = 1 \implies b = 1

; For factor

5

:

\min(c, 1) = 1 \implies c \ge 1

.

The GCD of two numbers takes the minimum exponent for each prime factor.

3

Apply the LCM exponent rule

\max(\text{exp}_N(p), \text{exp}_B(p)) = \text{exp}_{LCM}(p)

For factor

2

:

\max(a, 1) = 3 \implies a = 3

; For factor

5

:

\max(c, 2) = 3 \implies c = 3

. No prime factors greater than

5

exist in

N

.

The LCM of two numbers takes the maximum exponent for each prime factor.

4

Synthesize the exponents and compute

N

N = 2^3 \times 3^1 \times 5^3 = 8 \times 3 \times 125 = 3000

.

Multiplying out the uniquely determined prime factors gives the value of

N

.

Anahtar Kavram

Simultaneous prime exponent analysis using GCD (minimum exponents) and LCM (maximum exponents) rules.

Soru 634Soru →

A retail warehouse received a large shipment of electronics. On Monday, the warehouse sold $\frac{2}{5}$ of the total shipment. On Tuesday, it sold $25\%$ of the items that remained after Monday. On Wednesday, it sold $33\frac{1}{3}\%$ of the items that remained after Tuesday. If $120$ items remained unsold at the end of Wednesday, how many items were in the original shipment?

400

600

1,200

7,200

36

Cevabı ve açıklamayı göster

Cevap: $400$

Cevap

The original shipment contained

400

items.

The correct option correctly evaluates the fraction remaining after each successive sale. After selling

\frac{2}{5}

on Monday,

\frac{3}{5}

remained. After selling

25\%

(

\frac{1}{4}

) on Tuesday,

\frac{3}{4}

of the previous remainder (

\frac{9}{20}

) remained. After selling

33\frac{1}{3}\%

(

\frac{1}{3}

) on Wednesday,

\frac{2}{3}

of that remainder remained, giving a net remaining fraction of

\frac{3}{10}

. Equating

\frac{3}{10}

of the total to

120

yields an original shipment size of

400

.

Adım Adım Çözüm

1

Calculate the fraction of items remaining after Monday's sales.

Since

\frac{2}{5}

were sold,

1 - \frac{2}{5} = \frac{3}{5}

of the original shipment remained.

The remaining fraction is

1

minus the fraction sold.

2

Calculate the fraction of items remaining after Tuesday's sales.

Tuesday sold

25\% = \frac{1}{4}

of the remaining items, leaving

1 - \frac{1}{4} = \frac{3}{4}

of Monday's remainder. The fraction remaining relative to the original total is

\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}

.

Successive percentage reductions apply to the updated intermediate remainder.

3

Calculate the fraction of items remaining after Wednesday's sales.

Wednesday sold

33\frac{1}{3}\% = \frac{1}{3}

of Tuesday's remainder, leaving

1 - \frac{1}{3} = \frac{2}{3}

of that remainder. The final fraction remaining relative to the original total is

\frac{9}{20} \times \frac{2}{3} = \frac{6}{20} = \frac{3}{10}

.

Multiply by the fraction remaining after Wednesday's reduction.

4

Set up the equation to solve for the original total number of items

N

.

\frac{3}{10}N = 120 \implies N = 120 \times \frac{10}{3} = 400

.

Divide the remaining count by the net remaining fraction.

Anahtar Kavram

Successive Percent and Fraction Reductions

Tahmini Süre:1m 30s

Soru 635Soru →

For two positive integers $x$ and $y$ with $x < y$ , the greatest common divisor is $\text{GCD}(x, y) = 15$ and the least common multiple is $\text{LCM}(x, y) = 9000$ . If $x$ is a multiple of $8$ but not a multiple of $9$ , what is the value of $y - x$ ?

Cevabı ve açıklamayı göster

Cevap: 1005

Cevap

1005

By writing

x = 15a

and

y = 15b

with

\text{GCD}(a, b) = 1

, we derive

a \cdot b = \frac{9000}{15} = 600

. Prime factorizing

600 = 8 \times 3 \times 25

yields four coprime pairs

(a, b)

with

a < b

:

(1, 600)

,

(3, 200)

,

(8, 75)

, and

(24, 25)

. These yield candidate values for

x

of

15

,

45

,

120

, and

360

, respectively. Checking the divisibility conditions,

15

and

45

are not multiples of

8

, while

360

is a multiple of

9

. The only value of

x

that is a multiple of

8

and not a multiple of

9

is

120

(corresponding to

a = 8, b = 75

). Hence

y = 15 \times 75 = 1125

, and

y - x = 1125 - 120 = 1005

.

Adım Adım Çözüm

1

Set up the algebraic representation using the GCD

x = 15a

and

y = 15b

, where

\text{GCD}(a, b) = 1

and

a < b

Any two integers can be written as the product of their GCD and coprime factor multipliers.

2

Relate LCM and GCD to find the product of multipliers

a \times b

a \times b = \frac{\text{LCM}(x, y)}{\text{GCD}(x, y)} = \frac{9000}{15} = 600

The product of GCD and LCM equals the product of the numbers:

\text{GCD}(x,y) \times \text{LCM}(x,y) = x \times y = 15a \times 15b = 225ab

.

3

Decompose 600 into coprime component blocks

600 = 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25

Since

\text{GCD}(a, b) = 1

, prime powers cannot be split between

a

and

b

.

4

Form all valid candidate pairs

(a, b)

with

a < b

(1, 600)

,

(3, 200)

,

(8, 75)

, and

(24, 25)

There are

2^{3-1} = 4

ways to partition the 3 prime factor blocks into two coprime factors where

a < b

.

5

Apply divisibility constraints to isolate

x

and

y

x = 120

and

y = 1125

Only

x = 15 \times 8 = 120

satisfies being a multiple of 8 without being a multiple of 9.

6

Calculate the target difference

y - x

1125 - 120 = 1005

Subtracting

x

from

y

yields the required value.

Anahtar Kavram

Partitioning prime factor powers of LCM/GCD to identify coprime factor multipliers

Soru 636Soru →

If $n = 2^4 \times 5^3 \times 7^1$ , how many positive integer divisors of $n$ are divisible by $10$ but are not multiples of $35$ ?

8

12

15

20

24

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The value 12 is correct because a divisor

2^a \times 5^b \times 7^c

of

2^4 \times 5^3 \times 7^1

is divisible by 10 when

a \in \{1, 2, 3, 4\}

(4 options) and

b \in \{1, 2, 3\}

(3 options). To ensure it is not divisible by 35 (

5 \times 7

),

c

cannot be 1 when

b \ge 1

. Thus

c

must be 0 (1 option). Multiplying the choices yields

4 \times 3 \times 1 = 12

.

Adım Adım Çözüm

1

Express the general form of a divisor of

n

.

Any divisor

d

of

n = 2^4 \times 5^3 \times 7^1

can be written as

d = 2^a \times 5^b \times 7^c

, where

0 \le a \le 4

,

0 \le b \le 3

, and

0 \le c \le 1

.

Prime factorization determines the structure of all positive divisors.

2

Apply the condition that

d

must be divisible by

10

.

Since

10 = 2^1 \times 5^1

, we require

a \ge 1

and

b \ge 1

.

A number is divisible by 10 if and only if it contains at least one factor of 2 and at least one factor of 5.

3

Apply the condition that

d

must NOT be a multiple of

35

.

Since

35 = 5^1 \times 7^1

, a multiple of 35 must have

b \ge 1

and

c \ge 1

. Since

b \ge 1

is already satisfied, to avoid being a multiple of 35, we must set

c = 0

.

If

c = 1

, then combined with

b \ge 1

, the divisor would automatically be a multiple of 35.

4

Calculate the total number of valid choices for

(a, b, c)

.

a \in \{1, 2, 3, 4\}

(4 options),

b \in \{1, 2, 3\}

(3 options),

c \in \{0\}

(1 option). Total =

4 \times 3 \times 1 = 12

.

By the Fundamental Counting Principle, multiplying the independent choices gives the number of valid divisors.

Anahtar Kavram

Counting Divisors with Prime Factor Constraints

Tahmini Süre:1m 30s

Soru 637Soru →

A positive integer $n$ is not divisible by $3$ . If $n$ has exactly $15$ positive divisors and $2n$ has exactly $20$ positive divisors, how many positive divisors does $3n$ have?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

The number of positive divisors of an integer with prime factorization

p_1^{e_1} p_2^{e_2} \dots

is

(e_1 + 1)(e_2 + 1) \dots

. Given

d(n) = 15

,

n

can be

p^{14}

or

p^4 q^2

. Given

d(2n) = 20

, 2 must be a prime factor of

n

with exponent 2, making

n = 2^2 p^4

. Since

n

is not divisible by 3,

p \neq 3

. Multiplying

n

by 3 introduces

3^1

into the prime factorization, giving

3n = 2^2 \cdot 3^1 \cdot p^4

. The total number of positive divisors is

(2+1)(1+1)(4+1) = 30

.

Adım Adım Çözüm

1

Analyze the number of positive divisors of

n

.

n

is either

p^{14}

or

p^4 q^2

for distinct primes

p

and

q

.

The number of positive divisors of

n = p_1^{e_1} p_2^{e_2} \dots

is given by

(e_1 + 1)(e_2 + 1) \dots

. Since

15 = 15 \times 1 = 5 \times 3

, the exponent structure must be 14 or 4 and 2.

2

Analyze the number of positive divisors of

2n

.

n = 2^2 p^4

, where

p

is a prime other than 2 and 3.

If

n = p^4 q^2

, multiplying by 2 increases the exponent of 2 by 1. If

q = 2

, then

n = 2^2 p^4

and

2n = 2^3 p^4

, yielding

(3 + 1)(4 + 1) = 20

divisors. Any other prime structure yields a different count.

3

Determine the prime factorization of

3n

and count its positive divisors.

The number of divisors of

3n

is 30.

Since

n

is not divisible by 3,

p \neq 3

. Thus

3n = 2^2 \cdot 3^1 \cdot p^4

, which has

(2 + 1)(1 + 1)(4 + 1) = 30

positive divisors.

Anahtar Kavram

Divisor Count Formula and Prime Factorization

Soru 638Soru →

Let $K = 2^a \cdot 3^4 \cdot 7^b$ , where $a$ and $b$ are positive integers. If $K$ has exactly $48$ positive integer divisors that are multiples of $6$ , and exactly $30$ positive integer divisors that are multiples of $14$ , what is the total number of positive integer divisors of $K$ ?

42

56

60

70

84

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The total number of positive integer divisors of

K

is 70.

The prime factorization of

K

is

2^a \cdot 3^4 \cdot 7^b

. Any positive divisor of

K

takes the form

2^x \cdot 3^y \cdot 7^z

, where

0 \le x \le a

,

0 \le y \le 4

, and

0 \le z \le b

. A divisor is a multiple of

6 = 2^1 \cdot 3^1

if

x \ge 1

,

y \ge 1

, and

z \ge 0

. The number of such divisors is

a \times 4 \times (b+1) = 48

, which simplifies to

a(b+1) = 12

. A divisor is a multiple of

14 = 2^1 \cdot 7^1

if

x \ge 1

,

y \ge 0

, and

z \ge 1

. The number of such divisors is

a \times 5 \times b = 30

, which simplifies to

ab = 6

. Substituting

ab = 6

into

ab + a = 12

gives

6 + a = 12 \implies a = 6

, which means

b = 1

. The total number of positive integer divisors of

K = 2^6 \cdot 3^4 \cdot 7^1

is

(6+1)(4+1)(1+1) = 7 \times 5 \times 2 = 70

.

Adım Adım Çözüm

1

Express the condition for divisors of

K

being multiples of 6 in terms of exponents.

a \cdot 4 \cdot (b+1) = 48 \implies a(b+1) = 12

A divisor of

K = 2^a \cdot 3^4 \cdot 7^b

has the form

2^x \cdot 3^y \cdot 7^z

with

0 \le x \le a

,

0 \le y \le 4

,

0 \le z \le b

. For it to be a multiple of

6 = 2^1 \cdot 3^1

, we must have

x \ge 1

(

a

choices),

y \ge 1

(

4

choices), and

z \ge 0

(

b+1

choices).

2

Express the condition for divisors of

K

being multiples of 14 in terms of exponents.

a \cdot 5 \cdot b = 30 \implies ab = 6

For a divisor to be a multiple of

14 = 2^1 \cdot 7^1

, we must have

x \ge 1

(

a

choices),

y \ge 0

(

5

choices), and

z \ge 1

(

b

choices).

3

Solve the system of equations for the positive integer exponents

a

and

b

.

a = 6

and

b = 1

Expanding

a(b+1) = 12

gives

ab + a = 12

. Substituting

ab = 6

yields

6 + a = 12 \implies a = 6

. Then

6b = 6 \implies b = 1

.

4

Calculate the total number of positive integer divisors of

K = 2^6 \cdot 3^4 \cdot 7^1

.

(6+1)(4+1)(1+1) = 7 \times 5 \times 2 = 70

The total number of divisors of a prime factorization

p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is given by

(e_1 + 1)(e_2 + 1) \dots (e_k + 1)

.

Anahtar Kavram

Counting Divisors using Prime Factor Exponents

Tahmini Süre:2m 0s

Soru 639Soru →

A store owner purchases a batch of books for $\$150$ each and sells them at a price that is $40\%$ higher than the purchase price. During a clearance sale, the selling price is reduced by $10\%$ . What is the final sale price, in dollars, of one book?

Cevabı ve açıklamayı göster

Cevap: 189

Cevap

The final sale price of one book is

\$189

.

First, calculate the price after the

40\%

markup:

\$150 \times 1.40 = \$210

. Then apply the

10\%

discount to this new amount:

\$210 \times 0.90 = \$189

. The final price is

\$189

.

Adım Adım Çözüm

1

Calculate the price after a 40% markup on the base price of $150.

$210

A 40% markup means multiplying the original purchase price by 1.40.

2

Apply the 10% clearance discount to the marked-up price of $210.

$189

A 10% discount means multiplying the new price by 0.90.

Anahtar Kavram

Successive Percentage Change

Tahmini Süre:45s

Soru 640Soru →

At a technology firm, $\frac{3}{8}$ of the total annual operating budget was initially allocated to Research & Development, $0.35$ of the budget was allocated to Marketing, and the remainder was allocated to Operations. Mid-year, the Marketing allocation was increased by $20\%$ of its initial value, and the Operations allocation was decreased by $40\%$ of its initial value, while the Research & Development allocation remained unchanged. By what net percentage did the firm's total annual operating budget change?

A decrease of

20\%

A decrease of

6\%

A decrease of

4\%

An increase of

4\%

A decrease of

15\%

Cevabı ve açıklamayı göster

Cevap: A decrease of $4\%$

Cevap

A net decrease of

4\%

Converting

\frac{3}{8}

to

37.5\%

leaves

27.5\%

for Operations (

100\% - 37.5\% - 35\%

). A

20\%

increase on the

35\%

Marketing budget adds

7\%

to the overall total (

0.20 \times 35\% = +7\%

). A

40\%

decrease on the

27.5\%

Operations budget reduces the overall total by

11\%

(

0.40 \times 27.5\% = -11\%

). Summing these changes gives

+7\% - 11\% = -4\%

, representing a net decrease of

4\%

.

Adım Adım Çözüm

1

Convert initial fractional and decimal allocations to percentages of the total budget

Research & Development allocation =

\frac{3}{8} = 37.5\%

; Marketing allocation =

0.35 = 35\%

Converting all components to a common percentage format enables straightforward comparison and arithmetic.

2

Calculate the initial percentage allocated to Operations

Operations allocation =

100\% - (37.5\% + 35\%) = 100\% - 72.5\% = 27.5\%

The remainder of the budget after R&D and Marketing constitutes the Operations department share.

3

Determine the net percentage change contributed by each department's adjustment

Marketing change =

+20\% \text{ of } 35\% = +7\%

of total budget; Operations change =

-40\% \text{ of } 27.5\% = -11\%

of total budget; R&D change =

0\%

Each departmental percentage change must be weighted by that department's portion of the overall budget.

4

Sum the net contributions to find the overall budget change

Net overall change =

+7\% - 11\% = -4\%

(a decrease of

4\%

)

Combining the positive and negative adjustments yields the overall net percentage change relative to the initial budget.

Anahtar Kavram

Weighted Percent Change across Mixed Fractional and Decimal Sub-components

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