All practice questions

What is the remainder when $5^{99}$ is divided by $13$ ?

Show answer & explanation

Answer: 8

Answer

The remainder when

5^{99}

is divided by

13

is

8

.

To find the remainder of

5^{99}

divided by

13

, find the repeating pattern of remainders for powers of

5

modulo

13

. The first four powers yield remainders

5, 12, 8,

and

1

. Since

5^4 \equiv 1 \pmod{13}

, the remainders repeat every

4

powers. Dividing the exponent

99

by

4

gives

99 = 4 \times 24 + 3

. The remainder of

3

indicates that

5^{99}

has the exact same remainder modulo

13

as

5^3

, which is

8

.

Step-by-Step Solution

1

Analyze the remainders of powers of 5 modulo 13 to identify the repeating period length.

5^1 \equiv 5 \pmod{13}

,

5^2 \equiv 12 \pmod{13}

,

5^3 \equiv 8 \pmod{13}

, and

5^4 \equiv 1 \pmod{13}

. The sequence of remainders repeats every 4 powers.

Integral powers modulo a positive integer exhibit periodic behavior.

2

Compute the remainder of the exponent 99 divided by the cycle length 4.

99 \div 4 = 24

with a remainder of

3

.

The exponent's remainder modulo the cycle length determines which element in the periodic cycle gives the equivalent value.

3

Evaluate the 3rd term in the remainder cycle.

The 3rd element in the sequence

(5, 12, 8, 1)

is

8

.

An exponent congruent to

3 \pmod 4

yields the same remainder as

5^3 \pmod{13}

.

Key Concept

Modular Arithmetic and Cyclicity of Powers

Question 662Question →

At a chemical refining plant, a raw liquid compound containing Substance X, Substance Y, and an inert solvent is processed in two sequential purification phases. Initially, Substance X accounts for $0.25$ of the total weight of the compound, and Substance Y accounts for $\frac{2}{5}$ of the remaining weight, with the inert solvent comprising the rest. In Phase 1, $20\%$ of Substance X and $30\%$ of Substance Y are removed, while all of the inert solvent is retained. In Phase 2, a certain percentage $p\%$ of the inert solvent present after Phase 1 is removed, while no other substances are removed. If Substance X represents exactly $40\%$ of the total weight of the compound remaining after Phase 2, what is the value of $p$ ?

Show answer & explanation

Answer: 80

Answer

The value of

p

is 80.

Assuming a total initial weight of 100 units, Substance X is 25 units and the remaining weight is 75 units. Substance Y is

\frac{2}{5}

of 75, which equals 30 units, leaving 45 units of inert solvent. After Phase 1, 20 units of Substance X and 21 units of Substance Y remain, along with the full 45 units of solvent. In Phase 2, Substance X (20 units) becomes

40\%

of the total mixture, making the final total weight

\frac{20}{0.40} = 50

units. Since Substance X and Substance Y together account for

20 + 21 = 41

units, the remaining solvent after Phase 2 must be

50 - 41 = 9

units. Reducing solvent from 45 units down to 9 units requires removing

\frac{45 - 9}{45} = \frac{36}{45} = 80\%

of the solvent. Thus,

p = 80

.

Step-by-Step Solution

1

Determine initial component amounts using decimal and fractional breakdown.

In a 100-unit mixture, Substance X = 25 units, Substance Y = 30 units, and Inert Solvent = 45 units.

Substance X is

0.25

of the total (

25

units). Of the remaining

75

units, Substance Y is

\frac{2}{5} \times 75 = 30

units. The rest (

75 - 30 = 45

units) is inert solvent.

2

Calculate remaining component amounts after Phase 1 percentage reductions.

Substance X = 20 units, Substance Y = 21 units, Inert Solvent = 45 units.

Removing

20\%

of Substance X leaves

25 \times 0.80 = 20

units. Removing

30\%

of Substance Y leaves

30 \times 0.70 = 21

units. No solvent is removed in Phase 1.

3

Determine the final total mixture weight using the final percentage of Substance X.

Final total mixture weight = 50 units.

Substance X (20 units) represents

40\%

(

0.40

) of the final mixture after Phase 2, so the total weight is

\frac{20}{0.40} = 50

units.

4

Formulate and solve the linear equation for

p

.

p = 80

.

The total weight is the sum of all remaining components:

20 + 21 + 45\left(1 - \frac{p}{100}\right) = 50

. Solving

41 + 45\left(1 - \frac{p}{100}\right) = 50

yields

45\left(1 - \frac{p}{100}\right) = 9

, so

1 - \frac{p}{100} = 0.20

, giving

p = 80

.

Key Concept

Multi-step percentage change, fractional remaining parts, and algebraic mixture equations

Question 663Question →

An inventory manager is organizing items into storage boxes. When the items are packed into boxes of $24$ , $36$ , or $45$ items each, there are always $11$ items remaining unpacked. If the total number of items is between $700$ and $1,000$ , how many items are there in total?

Show answer & explanation

Answer: 731

Answer

731

The number of items

N

satisfies

N \equiv 11 \pmod{24}

,

N \equiv 11 \pmod{36}

, and

N \equiv 11 \pmod{45}

. Consequently,

N - 11

is a multiple of

\text{LCM}(24, 36, 45)

. The prime factorizations are

24 = 2^3 \times 3

,

36 = 2^2 \times 3^2

, and

45 = 3^2 \times 5

. The LCM is

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

. Thus,

N = 360k + 11

for some positive integer

k

. Testing values of

k

within the range

700 < N < 1,000

yields

k = 2

, giving

N = 360(2) + 11 = 731

.

Step-by-Step Solution

1

Find the Least Common Multiple (LCM) of the divisors 24, 36, and 45.

LCM(24, 36, 45) = 360

Any number that leaves a constant remainder when divided by a set of numbers must be a constant remainder greater than a common multiple of those numbers.

2

Formulate the general equation for the total number of items N.

N = 360k + 11

Adding the common remainder 11 to any multiple of the LCM yields a valid total number of items.

3

Evaluate integer values of k to satisfy the constraint 700 < N < 1,000.

k = 2 gives N = 731

k = 1 gives 371, which is too small, and k = 3 gives 1,091, which exceeds 1,000.

Key Concept

Least Common Multiple (LCM) with Constant Remainder

Question 664Question →

At the beginning of a fiscal year, a technology company allocated its total operating budget among three divisions: Research & Development, Marketing, and Operations. Research & Development received $0.45$ of the total budget, Marketing received $\frac{1}{3}$ of the total budget, and Operations received the remaining portion. Mid-year, the company adjusted the budget by increasing the Research & Development allocation by $20\%$ of its initial amount and reducing the Marketing allocation by $\frac{1}{4}$ of its initial amount, while the Operations allocation remained unchanged. If these mid-year adjustments resulted in a net increase of $\$30,000$ in the company's total operating budget, what was the company's total operating budget at the beginning of the fiscal year?

$187,500

$600,000

$2,700,000

$3,600,000

$4,500,000

Show answer & explanation

Answer: $4,500,000

Answer

$4,500,000

The option specifying

4,500,000 is correct. The increase in the Research & Development division is

20\%

of

0.45 B

, which equals

0.09 B

(or

\frac{9}{100} B

). The reduction in Marketing is

\frac{1}{4}

of

\frac{1}{3} B

, which equals

\frac{1}{12} B

. Combining these using a common denominator gives a net increase of

\frac{27}{300} B - \frac{25}{300} B = \frac{2}{300} B = \frac{1}{150} B

. Setting

\frac{1}{150} B = \

30,000

yields

B = \$4,500,000

.

Step-by-Step Solution

1

Express initial allocations as fractions of total budget B

Research & Development initial share =

0.45 B = \frac{9}{20} B

; Marketing initial share =

\frac{1}{3} B

Converting decimals to fractions simplifies exact combined calculations.

2

Calculate the monetary change in each division relative to total budget B

R&D increase =

20\% \times 0.45 B = 0.20 \times 0.45 B = 0.09 B = \frac{9}{100} B

. Marketing decrease =

\frac{1}{4} \times \frac{1}{3} B = \frac{1}{12} B

. Operations change =

0

.

Each change applies specifically to the division's initial allocation, not the total budget.

3

Find the net change in the total budget by taking the difference between the increase and the decrease

Net change =

\left(\frac{9}{100} - \frac{1}{12}\right) B = \left(\frac{27}{300} - \frac{25}{300}\right) B = \frac{2}{300} B = \frac{1}{150} B

Determining the net fraction allows direct equation setup with the given dollar increase.

4

Solve for the initial total operating budget B using the given net increase of $30,000

\frac{1}{150} B = 30,000 \implies B = 30,000 \times 150 = 4,500,000

Multiplying both sides by 150 yields the original total budget.

Key Concept

Combining decimal and fraction arithmetic to determine net percentage changes across component allocations.

Alternative Method

Convert all components to a common denominator out of 300 parts initially. Total budget = 300 parts. R&D = 135 parts (0.45). Marketing = 100 parts (1/3). Operations = 65 parts. R&D increase = 20% of 135 = +27 parts. Marketing decrease = 1/4 of 100 = -25 parts. Net change = +2 parts =

30,000. Therefore 1 part =

15,000, and 300 parts = $4,500,000.

Estimated Time:1m 45s

Question 665Question →

Let $N = 19^{43} + 11^{43} - 6^{43}$ . What is the remainder when $N$ is divided by $7$ ?

Show answer & explanation

Answer: 3

Answer

3

Reducing each base modulo 7 simplifies the expression significantly:

19 \equiv -2 \pmod{7}

,

11 \equiv 4 \pmod{7}

, and

6 \equiv -1 \pmod{7}

. Utilizing the cyclicity period of 3 for powers of 2 modulo 7 (

2^3 \equiv 1 \pmod{7}

),

19^{43} \equiv -(2^{43}) \equiv -2 \equiv 5 \pmod{7}

. Similarly,

11^{43} \equiv 4^{43} = 2^{86} = (2^3)^{28} \times 4 \equiv 4 \pmod{7}

. Lastly,

6^{43} \equiv (-1)^{43} = -1 \pmod{7}

. Combining these yields

N \equiv 5 + 4 - (-1) = 10 \equiv 3 \pmod{7}

.

Step-by-Step Solution

1

Reduce each base modulo 7

19 \equiv -2 \pmod{7}

,

11 \equiv 4 \pmod{7}

, and

6 \equiv -1 \pmod{7}

Simplifying bases using modular arithmetic reduces large power evaluations to smaller equivalents.

2

Evaluate

19^{43} \pmod{7}

using cyclicity

19^{43} \equiv -2 \equiv 5 \pmod{7}

Since

2^3 \equiv 1 \pmod{7}

and

43 = 3(14) + 1

,

2^{43} \equiv 2 \pmod{7}

, so

(-2)^{43} = -(2^{43}) \equiv -2 \pmod{7}

.

3

Evaluate

11^{43} \pmod{7}

using powers of 2

11^{43} \equiv 4 \pmod{7}

11^{43} \equiv 4^{43} = 2^{86} = (2^3)^{28} \times 2^2 \equiv 1^{28} \times 4 = 4 \pmod{7}

.

4

Evaluate

6^{43} \pmod{7}

6^{43} \equiv -1 \pmod{7}

Any odd power of

-1

equals

-1

.

5

Combine all components to determine the final remainder

N \equiv 5 + 4 - (-1) = 10 \equiv 3 \pmod{7}

Summing the individual remainders modulo 7 gives a final positive remainder of 3.

Key Concept

Modular Arithmetic and Cyclicity of Powers

Question 666Question →

A positive integer $n$ is divisible by $20$ but is not divisible by $8$ . If $n$ has exactly $15$ positive divisors and $10n$ has exactly $24$ positive divisors, what is the value of $n$ ?

Show answer & explanation

Answer: 2500

Answer

2500

To determine

n

, analyze its prime factorization. Divisibility by

20 = 2^2 \cdot 5^1

requires that

n

contains

2

raised to at least the power of

2

and

5

raised to at least the power of

1

. The condition that

n

is not divisible by

8 = 2^3

restricts the exponent of

2

to exactly

2

. Thus,

n = 2^2 \cdot 5^b \cdot K'

, where

b \ge 1

and

K'

is a product of powers of distinct prime factors other than

2

and

5

. The number of positive divisors of

n

is

d(n) = (2+1)(b+1)K = 3(b+1)K = 15

, which simplifies to

(b+1)K = 5

. Because

b \ge 1

, we have

b+1 \ge 2

. Since

5

is a prime number, its only divisor greater than or equal to

2

is

5

. Thus,

b+1 = 5

, giving

b = 4

, and

K = 1

, which means

n

has no prime factors other than

2

and

5

. Therefore,

n = 2^2 \cdot 5^4 = 4 \cdot 625 = 2500

. We verify that

10n = 2^3 \cdot 5^5

has

(3+1)(5+1) = 24

positive divisors, confirming the solution.

Step-by-Step Solution

1

Determine the power of 2 in the prime factorization of

n

The exponent of

2

in

n

is exactly

2

Because

n

is a multiple of

20 = 2^2 \cdot 5

but not a multiple of

8 = 2^3

,

2^2

divides

n

but

2^3

does not.

2

Set up the divisor counting formula for

n

d(n) = (2+1)(b+1)K = 15 \implies (b+1)K = 5

, where

b \ge 1

is the exponent of

5

and

K

represents the product of terms from any additional prime factors

The total number of positive divisors of an integer

N = p_1^{a_1} p_2^{a_2} \dots

is given by

(a_1+1)(a_2+1)\dots

3

Solve for the exponents and prime factors of

n

b = 4

and

K = 1

, giving

n = 2^2 \cdot 5^4

Since

n

is divisible by

20

,

5

is a prime factor of

n

, so

b \ge 1

, which implies

b+1 \ge 2

. Since

5

is prime, its only factor greater than

1

is

5

, forcing

b+1 = 5

and

K = 1

.

4

Verify

d(10n)

and evaluate

n

10n = 2^3 \cdot 5^5

has

(3+1)(5+1) = 24

divisors, and

n = 4 \cdot 625 = 2500

Multiplying

n

by

10 = 2 \cdot 5

increases the exponent of

2

from

2

to

3

and the exponent of

5

from

4

to

5

.

Key Concept

Divisor Count Formula and Prime Factorization Constraints

Question 667Question →

Let $S = \sum_{k=1}^{82} (-1)^k \cdot k \cdot 7^k$ . What is the remainder when $-S$ is divided by $10$ ?

1

3

7

9

0

Show answer & explanation

Answer: 9

Answer

The remainder when

-S

is divided by

10

is

9

.

Grouping the 82 terms into 20 blocks of 4 shows that each block contributes

6 \pmod{10}

, summing to

120 \equiv 0 \pmod{10}

. The 81st and 82nd terms contribute

3

and

8

respectively, giving

S \equiv 11 \equiv 1 \pmod{10}

. Therefore,

-S \equiv -1 \equiv 9 \pmod{10}

.

Step-by-Step Solution

1

Analyze the units digit cyclicity of

7^k \pmod{10}

.

The sequence of units digits for

7^k

repeats every 4 powers:

7^1 \equiv 7

,

7^2 \equiv 9

,

7^3 \equiv 3

,

7^4 \equiv 1 \pmod{10}

.

Units digits of powers of 7 follow a repeating cycle of length 4.

2

Evaluate the terms modulo 10 for a general block of 4 consecutive integers

k = 4m+1, 4m+2, 4m+3, 4m+4

.

The four terms evaluate to:

(-1)^{4m+1}(4m+1)7^{4m+1} \equiv -(4m+1)(7) \equiv 2m+3 \pmod{10}

;

(-1)^{4m+2}(4m+2)7^{4m+2} \equiv (4m+2)(9) \equiv 6m+8 \pmod{10}

;

(-1)^{4m+3}(4m+3)7^{4m+3} \equiv -(4m+3)(3) \equiv 8m+1 \pmod{10}

; and

(-1)^{4m+4}(4m+4)7^{4m+4} \equiv (4m+4)(1) \equiv 4m+4 \pmod{10}

.

Modular arithmetic simplifies each term according to its position in the 4-term cycle.

3

Sum the four terms in a single block of 4.

(2m+3) + (6m+8) + (8m+1) + (4m+4) = 20m + 16 \equiv 6 \pmod{10}.

Every complete block of 4 consecutive terms contributes exactly 6 modulo 10, independent of

m

.

4

Calculate the total sum

S \pmod{10}

for

k=1

to

82

.

82 terms contain 20 full blocks of 4 terms (

k=1

to

80

) plus 2 remaining terms (

k=81

and

k=82

). The 20 blocks contribute

20 \times 6 = 120 \equiv 0 \pmod{10}

. For

m=20

,

k=81

gives

2(20)+3 = 43 \equiv 3 \pmod{10}

, and

k=82

gives

6(20)+8 = 128 \equiv 8 \pmod{10}

. Thus

S \equiv 0 + 3 + 8 = 11 \equiv 1 \pmod{10}

.

Summing the contributions of all complete blocks and the remaining boundary terms gives

S \pmod{10}

.

5

Determine the positive remainder of

-S

when divided by 10.

Since

S \equiv 1 \pmod{10}

,

-S \equiv -1 \equiv 10 - 1 = 9 \pmod{10}

.

By definition of integer division, the remainder

r

must satisfy

0 \le r < 10

.

Key Concept

Units Digit Cyclicity and Modular Arithmetic with Negative Integer Remainders

Question 668Question →

If $|2x - 3| = 7$ , which of the following is a possible value of $x$ ?

5

-5

2

10

-4

Show answer & explanation

Answer: $5$

Answer

The value

5

is a valid solution to the given absolute value equation.

The absolute value equation

|2x - 3| = 7

splits into two linear equations:

2x - 3 = 7

and

2x - 3 = -7

. Solving the first equation gives

2x = 10

, so

x = 5

. Solving the second gives

2x = -4

, so

x = -2

. Among the listed options,

5

is the correct solution.

Step-by-Step Solution

1

Set up the two cases for the absolute value equation

|2x - 3| = 7

.

Case 1:

2x - 3 = 7

; Case 2:

2x - 3 = -7

.

An absolute value expression

|u| = c

(where

c \geq 0

) resolves to

u = c

or

u = -c

.

2

Solve Case 1:

2x - 3 = 7

.

2x = 10 \implies x = 5

.

Add 3 to both sides and divide by 2.

3

Solve Case 2:

2x - 3 = -7

.

2x = -4 \implies x = -2

.

Add 3 to both sides and divide by 2.

4

Compare the solutions (

x = 5

and

x = -2

) with the given choices.

The value

5

is present among the options.

Identifies the correct choice matching one of the calculated solutions.

Key Concept

Solving Linear Absolute Value Equations

Estimated Time:45s

Question 669Question →

To reduce municipal carbon emissions and alleviate gridlock, City X introduced a daily congestion toll on private passenger automobiles entering its commercial core during morning peak hours. Analysts argue that this policy will successfully decrease overall motor vehicle traffic along the primary access corridors leading into the commercial core by at least 25 percent. Their reasoning is that the added financial cost will compel a significant portion of daily automobile commuters to switch to the city's underutilized suburban rail network. Which of the following, if true, most seriously weakens the analysts' argument?

A municipal survey conducted prior to the toll's implementation indicated that over 30 percent of private automobile commuters expressed a willingness to use public transit if driving costs increased.

Because the toll applies exclusively to private passenger automobiles, commercial delivery fleets and ride-hailing services plan to sharply increase peak-hour trips along those same access corridors to meet shifted consumer demand.

Suburban municipalities surrounding City X that enacted similar congestion pricing schemes five years ago observed a gradual rise in local residential property values near rail stations.

The net revenue collected from the congestion toll will be insufficient to fully offset the municipal government's projected annual budget deficit for road maintenance.

Standard public transit fares across City X are scheduled for a minor, routine inflation adjustment at the end of the upcoming fiscal year.

Show answer & explanation

Answer: Because the toll applies exclusively to private passenger automobiles, commercial delivery fleets and ride-hailing services plan to sharply increase peak-hour trips along those same access corridors to meet shifted consumer demand.

Answer

The argument is most seriously weakened by the statement that commercial delivery fleets and ride-hailing services will increase peak-hour trips along the access corridors, replacing reduced private cars with other motor vehicles.

The correct answer demonstrates that while private passenger cars may decrease due to the toll, another category of motor vehicles—commercial fleets and ride-hailing cars—will significantly increase operations on the exact same access corridors. This directly invalidates the analysts' assumption that reduced private automobile commuter traffic guarantees a 25 percent drop in total motor vehicle volume.

Step-by-Step Solution

1

Identify the main conclusion and premises.

Conclusion: Overall motor vehicle traffic along primary access corridors leading into the commercial core will decrease by at least 25%. Premise: A daily toll on private passenger cars will cause commuters to switch to suburban rail.

Deconstructing the argument isolates the leap between private car commuter behavior and total vehicle traffic volume.

2

Identify unstated assumptions in the argument's logic.

The argument assumes that private passenger vehicles are the sole major contributor to traffic volume on those corridors, or that decrease in private vehicles will not be replaced by other types of vehicular traffic.

Weakening questions require finding new information that shatters an unstated assumption.

3

Evaluate option impacts against the core assumption.

Showing that exempt vehicle categories (delivery fleets and ride-hailing cars) will increase trip frequency along the exact same corridors directly counteracts the projected net traffic reduction.

Demonstrating an offsetting surge in non-tolled motor vehicles directly breaks the link between fewer private cars and lower overall corridor traffic volume.

Key Concept

Evaluating Causal and Extrapolation Arguments by Identifying Offsetting Alternative Factors

Question 670Question →

Let $N = 2^3 \cdot 3^a \cdot 5^b$ , where $a$ and $b$ are positive integers. If $N$ has exactly 60 positive integer divisors that are multiples of 10 and exactly 40 positive integer divisors that are multiples of 15, how many positive integer divisors of $N$ are multiples of 12?

20

22

24

33

66

Show answer & explanation

Answer: 22

Answer

22

The correct answer is 22. From the problem statement, divisors divisible by 10 require

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

,

y \in \{0, \dots, a\}

, and

z \in \{1, \dots, b\}

, leading to

3(a+1)b = 60

, or

(a+1)b = 20

. Divisors divisible by 15 require

x \in \{0, 1, 2, 3\}

,

y \in \{1, \dots, a\}

, and

z \in \{1, \dots, b\}

, leading to

4ab = 40

, or

ab = 10

. Substituting

ab = 10

into

ab + b = 20

yields

b = 10

and

a = 1

. To be a multiple of

12 = 2^2 \cdot 3^1

, a divisor must have

x \in \{2, 3\}

(2 choices),

y \in \{1\}

(1 choice), and

z \in \{0, 1, \dots, 10\}

(11 choices). The total number of such divisors is

2 \times 1 \times 11 = 22

.

Step-by-Step Solution

1

Set up the equation for positive divisors of

N

that are multiples of 10.

Multiples of 10 must be divisible by

2^1 \cdot 5^1

. Thus, for a divisor

2^x \cdot 3^y \cdot 5^z

, we need

1 \le x \le 3

(3 choices),

0 \le y \le a

(

a+1

choices), and

1 \le z \le b

(

b

choices). The total number of such divisors is

3(a + 1)b = 60

, which simplifies to

(a + 1)b = 20

.

Divisibility by 10 requires at least one factor of 2 and at least one factor of 5.

2

Set up the equation for positive divisors of

N

that are multiples of 15.

Multiples of 15 must be divisible by

3^1 \cdot 5^1

. Thus, we need

0 \le x \le 3

(4 choices),

1 \le y \le a

(

a

choices), and

1 \le z \le b

(

b

choices). The total number of such divisors is

4ab = 40

, which simplifies to

ab = 10

.

Divisibility by 15 requires at least one factor of 3 and at least one factor of 5.

3

Solve the system of equations for

a

and

b

.

Expanding

(a + 1)b = 20

yields

ab + b = 20

. Substituting

ab = 10

into this equation gives

10 + b = 20 \implies b = 10

. Since

ab = 10

, we find

a(10) = 10 \implies a = 1

.

Determining the exact exponents

a

and

b

allows us to fully specify

N = 2^3 \cdot 3^1 \cdot 5^{10}

.

4

Calculate the number of divisors of

N

that are multiples of 12.

Multiples of 12 must be divisible by

12 = 2^2 \cdot 3^1

. For a divisor

2^x \cdot 3^y \cdot 5^z

, we require

2 \le x \le 3

(2 choices:

2^2, 2^3

),

1 \le y \le a = 1

(1 choice:

3^1

), and

0 \le z \le b = 10

(

b + 1 = 11

choices:

5^0, 5^1, \dots, 5^{10}

). Total multiples of 12 =

2 \times 1 \times 11 = 22

.

Applying the fundamental counting principle to the constrained prime factor exponent ranges.

Key Concept

Divisor counting formula under prime factor constraint conditions

Question 671Question →

A 6-letter security code is created by arranging all the letters in the word $BANANA$ . How many distinct security codes can be formed such that the two letters $N$ are not adjacent to each other?

20

30

40

60

120

Show answer & explanation

Answer: 40

Answer

40 distinct security codes can be formed such that the two

N

's are not adjacent.

To find the number of arrangements where the two

N

's are not adjacent, use complementary counting. First, compute total unrestricted permutations of

BANANA

(1

B

, 3

A

's, 2

N

's), which gives

\frac{6!}{3! \times 2!} = 60

. Next, count the forbidden arrangements where the two

N

's are together by grouping them into a single block

(NN)

. Arranging

B

,

A

,

A

,

A

, and

(NN)

yields

\frac{5!}{3!} = 20

. Subtracting forbidden arrangements from total arrangements gives

60 - 20 = 40

.

Step-by-Step Solution

1

Calculate the total number of distinct arrangements of the letters in

BANANA

without restrictions.

The word

BANANA

contains 6 letters in total: 1

B

, 3

A

's, and 2

N

's. The number of distinct permutations is

\frac{6!}{1! \times 3! \times 2!} = \frac{720}{1 \times 6 \times 2} = 60

.

Indistinguishable duplicate letters must be accounted for by dividing the total permutations by the factorial of each letter's count.

2

Calculate the number of arrangements where the two

N

's ARE adjacent.

Treat the pair

NN

as a single block. The items to arrange are

B

,

A

,

A

,

A

, and

(NN)

, which gives 5 items with 3 identical

A

's. The number of arrangements is

\frac{5!}{1! \times 3! \times 1!} = \frac{120}{6} = 20

.

Grouping restricted elements into a single composite element simplifies calculating forbidden configurations.

3

Apply complementary counting to find the number of valid arrangements where the two

N

's are not adjacent.

Total valid arrangements = Total arrangements - Adjacent arrangements =

60 - 20 = 40

.

Complementary counting is the most efficient method when calculating non-adjacent placements.

Key Concept

Counting with Restrictions and Repetitions (Complementary Counting Principle)

Estimated Time:1m 30s

Question 672Question →

A boutique investment fund allocates its capital among Technology, Healthcare, and Renewable Energy in the ratio $3 : 4 : 5$ , respectively. After receiving an additional $\$12$ million in new capital, the entire new amount is invested in Healthcare, changing the ratio of capital among Technology, Healthcare, and Renewable Energy to $3 : 6 : 5$ . What was the total initial capital, in millions of dollars, invested in the fund?

24

72

84

36

60

Show answer & explanation

Answer: $72$

Answer

72

million dollars

Let the initial capital allocations be

3x

,

4x

, and

5x

million dollars for Technology, Healthcare, and Renewable Energy, respectively. The initial total capital is

3x + 4x + 5x = 12x

. When

\$12

million is added to Healthcare, its new amount becomes

4x + 12

. Comparing the unchanged Technology amount to the updated Healthcare amount gives the proportion

\frac{3x}{4x + 12} = \frac{3}{6} = \frac{1}{2}

. Cross-multiplying yields

6x = 4x + 12

, so

2x = 12

and

x = 6

. The total initial capital is

12 \times 6 = 72

million dollars.

Step-by-Step Solution

1

Represent initial allocations using a multiplier

x

Technology

= 3x

, Healthcare

= 4x

, Renewable Energy

= 5x

Ratios represent proportional parts, so multiplying each term by a common constant

x

expresses the actual amounts.

2

Set up the equation following the addition of

\$12

million to Healthcare

New Healthcare allocation

= 4x + 12

. The new ratio is

3x : (4x + 12) : 5x = 3 : 6 : 5

.

Only the Healthcare amount changes, while Technology and Renewable Energy allocations remain

3x

and

5x

respectively.

3

Solve for

x

using the ratio between Technology and Healthcare

\frac{3x}{4x + 12} = \frac{3}{6} \implies \frac{3x}{4x + 12} = \frac{1}{2} \implies 6x = 4x + 12 \implies 2x = 12 \implies x = 6

Equating the algebraic ratio of Technology to Healthcare to the new simplified ratio allows solving for

x

.

4

Calculate the total initial capital

\text{Total initial capital} = 3x + 4x + 5x = 12x = 12(6) = 72

Summing the initial parts (

12x

) and substituting

x = 6

yields the initial total.

Key Concept

Ratio scaling and constant-part proportion adjustments

Estimated Time:1m 30s

Question 673Question →

How many distinct 6-digit positive integers can be formed using all of the digits $1, 1, 2, 2, 3,$ and $4$ such that no two identical digits are adjacent to each other?

36

60

72

84

96

Show answer & explanation

Answer: 84

Answer

84

To find the number of 6-digit integers where no two identical digits are adjacent, apply complementary counting with the Principle of Inclusion-Exclusion. First, calculate the total distinct arrangements of the digits

1, 1, 2, 2, 3, 4

, which is

\frac{6!}{2!2!} = 180

. Next, count the forbidden outcomes. Treating the two 1s as a single block gives

\frac{5!}{2!} = 60

arrangements with adjacent 1s. Similarly, treating the two 2s as a single block gives

\frac{5!}{2!} = 60

arrangements with adjacent 2s. Arrangements where both the 1s are adjacent and the 2s are adjacent use two single blocks along with digits 3 and 4, giving

4! = 24

arrangements. By Inclusion-Exclusion, the total number of forbidden arrangements is

60 + 60 - 24 = 96

. Subtracting these from the total unrestricted arrangements gives

180 - 96 = 84

.

Step-by-Step Solution

1

Calculate the total number of unrestricted distinct 6-digit arrangements of the multiset {1, 1, 2, 2, 3, 4}.

Total arrangements =

\frac{6!}{2! \times 2!} = \frac{720}{4} = 180

.

Dividing by

2! \times 2!

accounts for the indistinguishable repeated digits 1 and 2.

2

Define the forbidden events and calculate their individual counts.

Let

A

be the set of arrangements with adjacent 1s, and

B

be the set of arrangements with adjacent 2s.
For

|A|

, treat

(11)

as a single block: elements are

\{(11), 2, 2, 3, 4\}

, so

|A| = \frac{5!}{2!} = 60

.
For

|B|

, treat

(22)

as a single block: elements are

\{1, 1, (22), 3, 4\}

, so

|B| = \frac{5!}{2!} = 60

.

Grouping adjacent identical numbers into a single block simplifies counting restriction violations.

3

Calculate the intersection count

|A \cap B|

where both pairs are adjacent.

Treating both

(11)

and

(22)

as single blocks yields elements

\{(11), (22), 3, 4\}

, so

|A \cap B| = 4! = 24

.

Arrangements where both pairs of identical digits are adjacent are counted in both

A

and

B

and must be identified.

4

Apply the Principle of Inclusion-Exclusion to find the total forbidden arrangements and subtract from the total unrestricted arrangements.

|A \cup B| = |A| + |B| - |A \cap B| = 60 + 60 - 24 = 96

.
Valid arrangements =

180 - 96 = 84

.

Complementary counting yields the exact number of arrangements where no two identical digits are adjacent.

Key Concept

Permutations of Multisets with Restrictions via Inclusion-Exclusion

Question 674Question →

A 7-digit security passcode is to be formed using the digits $1, 1, 2, 2, 3, 4,$ and $5$ . How many distinct 7-digit passcodes can be formed such that neither the two $1$ s nor the two $2$ s are adjacent to each other?

540

660

780

900

1,020

Show answer & explanation

Answer: 660

Answer

660 distinct passcodes can be formed.

To find the number of arrangements where neither identical pair is adjacent, we use complementary counting. First, compute total unrestricted arrangements of the digits

1,1,2,2,3,4,5

, which is

\frac{7!}{2!2!} = 1,260

. Next, by treating

(11)

as a single block, there are

\frac{6!}{2!} = 360

arrangements with adjacent

1

s. Similarly, treating

(22)

as a single block yields

\frac{6!}{2!} = 360

arrangements with adjacent

2

s. Arrangements where both pairs are adjacent treat

(11)

and

(22)

as two single blocks alongside

3,4,5

, giving

5! = 120

arrangements. Applying Inclusion-Exclusion gives

360 + 360 - 120 = 600

forbidden arrangements. Subtracting from the total gives

1,260 - 600 = 660

.

Step-by-Step Solution

1

Calculate the total number of distinct arrangements without any restrictions.

Total arrangements =

\frac{7!}{2! \times 2!} = \frac{5,040}{4} = 1,260

.

There are 7 digits total, with the digit 1 repeated twice and the digit 2 repeated twice.

2

Calculate the number of arrangements where the two

1

s are adjacent.

N(\text{adjacent } 1\text{s}) = \frac{6!}{2!} = \frac{720}{2} = 360

.

Treat the pair

(11)

as a single unit, leaving 6 items:

(11), 2, 2, 3, 4, 5

with two identical

2

s.

3

Calculate the number of arrangements where the two

2

s are adjacent.

N(\text{adjacent } 2\text{s}) = \frac{6!}{2!} = \frac{720}{2} = 360

.

Treat the pair

(22)

as a single unit, leaving 6 items:

1, 1, (22), 3, 4, 5

with two identical

1

s.

4

Calculate the number of arrangements where both the two

1

s and two

2

s are simultaneously adjacent.

N(\text{both adjacent}) = 5! = 120

.

Treat

(11)

and

(22)

each as single distinct units, giving 5 distinct items:

(11), (22), 3, 4, 5

.

5

Apply the Principle of Inclusion-Exclusion to find the total restricted count.

Valid passcodes =

1,260 - (360 + 360 - 120) = 1,260 - 600 = 660

.

Subtract the union of forbidden cases from the total arrangements:

N(\text{total}) - [N(\text{1s adj}) + N(\text{2s adj}) - N(\text{both adj})]

.

Key Concept

Complementary Counting with Inclusion-Exclusion Principle on Permutations of Repeated Elements

Estimated Time:2m 0s

Question 675Question →

For all real numbers $a$ and $b$ , the custom operation $\star$ is defined by $a \star b = (a + b)^2 - 3b$ . What is the value of $4 \star 2$ ?

30

24

14

33

18

Show answer & explanation

Answer: 30

Answer

30

Substituting

a = 4

and

b = 2

into the definition

(a + b)^2 - 3b

gives

(4 + 2)^2 - 3(2) = 6^2 - 6 = 36 - 6 = 30

.

Step-by-Step Solution

1

Identify the values assigned to the custom operator variables

a = 4

and

b = 2

The expression

4 \star 2

corresponds to evaluating

a \star b

where

a

is the first operand and

b

is the second operand.

2

Substitute

a = 4

and

b = 2

into the operator definition

(a + b)^2 - 3b

(4 + 2)^2 - 3(2)

Direct substitution into the given algebraic definition.

3

Perform the operations in parentheses and exponentiation

6^2 - 6 = 36 - 6

First add inside the parentheses (

4 + 2 = 6

), then square the sum (

6^2 = 36

), and multiply

3 \times 2 = 6

.

4

Subtract to find the final value

30

Subtracting

6

from

36

yields

30

.

Key Concept

Custom Operator Evaluation

Estimated Time:45s

Question 676Question →

Three industrial machines, $A$ , $B$ , and $C$ , operating independently at their respective constant rates, can complete a production order of $1,200$ units in $12$ hours, $15$ hours, and $20$ hours, respectively. All three machines start working simultaneously on the order at 8:00 AM. At 10:00 AM, Machine $A$ experiences a mechanical failure and stops working permanently. Machine $B$ and Machine $C$ continue working together until 11:00 AM, at which point Machine $B$ 's operating efficiency drops by $50\%$ due to overheating, while Machine $C$ continues at its original rate. At what time will the production order of $1,200$ units be fully completed?

4:48 PM

3:09 PM

5:12 PM

4:18 PM

6:30 PM

Show answer & explanation

Answer: 4:48 PM

Answer

The production order will be completed at 4:48 PM.

The correct answer of 4:48 PM is derived by dividing the job into three distinct stages. From 8:00 AM to 10:00 AM (2 hours), Machines A, B, and C produce 240 units/hour, yielding 480 units. From 10:00 AM to 11:00 AM (1 hour), Machines B and C produce 140 units/hour, yielding 140 units, leaving 580 units remaining. After 11:00 AM, Machine B operates at 40 units/hour and Machine C at 60 units/hour, producing a combined rate of 100 units/hour. Dividing 580 remaining units by 100 units/hour gives 5.8 hours (5 hours 48 minutes), placing completion at exactly 4:48 PM.

Step-by-Step Solution

1

Determine individual production rates for each machine.

Machine

A

produces

\frac{1,200}{12} = 100

units/hour. Machine

B

produces

\frac{1,200}{15} = 80

units/hour. Machine

C

produces

\frac{1,200}{20} = 60

units/hour.

Establishing hourly rates is necessary to calculate work completed in each time interval.

2

Calculate work completed during Stage 1 (8:00 AM to 10:00 AM).

Combined rate of

A + B + C = 100 + 80 + 60 = 240

units/hour. In

2

hours, work completed

= 240 \times 2 = 480

units. Remaining units

= 1,200 - 480 = 720

units.

All three machines work together for 2 full hours before Machine A breaks down.

3

Calculate work completed during Stage 2 (10:00 AM to 11:00 AM).

Combined rate of

B + C = 80 + 60 = 140

units/hour. In

1

hour, work completed

= 140 \times 1 = 140

units. Remaining units

= 720 - 140 = 580

units.

Machine A is inactive, leaving only Machines B and C operating at full capacity for 1 hour.

4

Calculate the time required for Stage 3 (from 11:00 AM until completion).

Machine

B

's reduced rate

= 80 \times 0.50 = 40

units/hour. New combined rate of

B + C = 40 + 60 = 100

units/hour. Time needed

= \frac{580}{100} = 5.8

hours

= 5

hours and

48

minutes.

Machine B operates at half speed while Machine C continues at full speed to finish the remaining 580 units.

5

Determine the final completion clock time.

11:00 AM

+ 5

hours and

48

minutes

= 4:48

PM.

Adding the duration of the final stage to the starting time of 11:00 AM gives the exact completion time.

Key Concept

Multi-stage combined work problems require calculating individual rates, tracking partial work completed in each phase, and adjusting combined rates whenever active entities or their individual rates change.

Estimated Time:3m 0s

Question 677Question →

An investment fund initially divides its total capital among three portfolios—Real Estate, Stocks, and Bonds—in the ratio $3 : 4 : 5$ . During a financial restructuring, a portion of capital is transferred directly from the Bonds portfolio to the Real Estate portfolio while the capital in Stocks remains unchanged, resulting in a new ratio of Real Estate to Stocks of $5 : 4$ . Subsequently, an additional $\$60,000$ in external capital is deposited into the Bonds portfolio, making the ratio of Stocks to Bonds $2 : 3$ . What was the total initial capital in the fund across all three portfolios?

$180,000

$200,000

$240,000

$300,000

$720,000

Show answer & explanation

Answer: $240,000

Answer

The total initial capital in the fund was $240,000.

The correct answer of

240,000 is obtained by tracking each portfolio's balance across the two sequential changes. Initially, Real Estate, Stocks, and Bonds are

3x

,

4x

, and

5x

respectively (total

12x

). When Real Estate increases to

5x

relative to Stocks (

4x

), the required capital comes entirely from Bonds, reducing Bonds from

5x

to

3x

. Adding

\

60,000

to Bonds gives a new balance of

3x + 60,000

. Establishing the new Stocks to Bonds ratio

\frac{4x}{3x + 60,000} = \frac{2}{3}

yields

x = 20,000

. Multiplying by the initial total of 12 parts yields

12 \times 20,000 =

240,000$.

Step-by-Step Solution

1

Express initial portfolio amounts in terms of a multiplier

x

Real Estate =

3x

, Stocks =

4x

, Bonds =

5x

. Total initial capital =

3x + 4x + 5x = 12x

.

The given initial ratio of Real Estate : Stocks : Bonds is

3 : 4 : 5

.

2

Account for the internal capital transfer between Bonds and Real Estate

Real Estate becomes

5x

, Stocks remains

4x

, and Bonds becomes

3x

.

Stocks remains

4x

. Since the ratio of Real Estate to Stocks becomes

5 : 4

, Real Estate must equal

5x

. Because capital was only moved internally from Bonds to Real Estate, the sum of Real Estate and Bonds remains

3x + 5x = 8x

, leaving Bonds with

8x - 5x = 3x

.

3

Set up an algebraic proportion after adding

\$60,000

to Bonds

\frac{4x}{3x + 60,000} = \frac{2}{3}

The new ratio of Stocks to Bonds is given as

2 : 3

.

4

Solve for the multiplier

x

and determine the total initial capital

3(4x) = 2(3x + 60,000) \Rightarrow 12x = 6x + 120,000 \Rightarrow 6x = 120,000 \Rightarrow x = 20,000

. Total initial capital

= 12x = 12 \times 20,000 =

240,000$.

Cross-multiplying yields the value of

x

, which is then scaled by the total initial ratio sum of 12 parts.

Key Concept

Multi-Step Ratio Adjustments and Internal vs. External Capital Changes

Estimated Time:2m 0s

Question 678Question →

Company A charges a one-time fixed fee of $\$20$ plus $\$5$ per hour to rent a piece of machinery. Company B charges a one-time fixed fee of $\$35$ plus $\$2$ per hour to rent the same machinery. For how many hours of rental will the total cost charged by both companies be equal?

3

5

7

11

15

Show answer & explanation

Answer: 5

Answer

The total cost at both companies will be equal for a rental duration of 5 hours.

The cost charged by Company A is modeled by

20 + 5h

and Company B by

35 + 2h

. Setting these equal yields

20 + 5h = 35 + 2h

, which simplifies to

3h = 15

, giving

h = 5

. Thus, 5 hours is the correct rental duration.

Step-by-Step Solution

1

Define the variable and write total cost equations for each company.

Let

h

represent the number of hours. Company A's cost is

C_A = 20 + 5h

and Company B's cost is

C_B = 35 + 2h

.

Linear cost models combine a fixed base cost with a variable hourly component.

2

Set the two cost equations equal to solve for

h

.

20 + 5h = 35 + 2h

The total costs are equal when

C_A = C_B

.

3

Isolate the variable term

h

on one side of the equation.

5h - 2h = 35 - 20 \Rightarrow 3h = 15 \Rightarrow h = 5

Subtracting

2h

and

20

from both sides simplifies the equation to solve directly for

h

.

Key Concept

Setting up and solving single-variable linear equations from real-world cost contexts.

Estimated Time:45s

Question 679Question →

If the quadratic equation $x^2 - kx + 36 = 0$ has two distinct positive integer roots, $r_1$ and $r_2$ , and $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 19x + m = 0$ , what is the value of $m$ ?

Show answer & explanation

Answer: 70

Answer

The value of

m

is 70.

By applying Vieta's formulas to both quadratic equations, we find that the sum of the original roots is

r_1 + r_2 = 15

and their product is

r_1 r_2 = 36

. Factoring

x^2 - 15x + 36 = (x-3)(x-12) = 0

confirms the original roots are

3

and

12

. The new roots are

3+2=5

and

12+2=14

. Their product

m = 5 \times 14 = 70

.

Step-by-Step Solution

1

Apply Vieta's relations to the first quadratic equation

r_1 + r_2 = k

and

r_1 \cdot r_2 = 36

For any quadratic equation

x^2 + bx + c = 0

, the sum of roots is

-b

and the product of roots is

c

.

2

Apply Vieta's relations to the second quadratic equation

(r_1 + 2) + (r_2 + 2) = 19 \implies r_1 + r_2 = 15

The coefficient of

x

in

x^2 - 19x + m = 0

dictates that the sum of its roots equals

19

.

3

Determine the roots

r_1

and

r_2

and check integer constraints

r_1 = 3

and

r_2 = 12

The integer factors of

36

that sum to

15

are

3

and

12

, satisfying all conditions.

4

Compute the constant term

m

for the transformed equation

m = (3 + 2)(12 + 2) = 5 \times 14 = 70

The constant term

m

equals the product of the transformed roots

(r_1 + 2)

and

(r_2 + 2)

.

Key Concept

Vieta's Formulas and Quadratic Root Transformations

Estimated Time:2m 0s

Question 680Question →

For all non-zero real numbers $a$ and $b$ , the custom operator $\Delta$ is defined by $a \Delta b = \frac{a^2 - b^2}{ab}$ . A function $f(x)$ is defined for all $x \neq 0$ and $x \neq -1$ by $f(x) = x \Delta (x + 1)$ . If $k$ is a positive constant such that $f(k) = -\frac{7}{12}$ , what is the value of $(k + 2) \Delta k$ ?

-\frac{16}{15}

\frac{4}{15}

\frac{5}{6}

\frac{16}{15}

\frac{3}{2}

Show answer & explanation

Answer: $\frac{16}{15}$

Answer

The value of

(k + 2) \Delta k

is

\frac{16}{15}

.

Applying the custom operator to

f(x)

yields

f(x) = \frac{x^2 - (x+1)^2}{x(x+1)} = \frac{-2x - 1}{x^2 + x}

. Setting

f(k) = -\frac{7}{12}

gives

\frac{2k + 1}{k^2 + k} = \frac{7}{12}

, which simplifies to

7k^2 - 17k - 12 = 0

. Factoring

(7k + 4)(k - 3) = 0

and selecting the positive root gives

k = 3

. Evaluating

(3+2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \cdot 3} = \frac{16}{15}

.

Step-by-Step Solution

1

Express

f(x)

using the definition of the custom operator

\Delta

.

f(x) = x \Delta (x + 1) = \frac{x^2 - (x + 1)^2}{x(x + 1)} = \frac{x^2 - (x^2 + 2x + 1)}{x(x + 1)} = \frac{-2x - 1}{x^2 + x}

Substitute

a = x

and

b = x + 1

into

a \Delta b = \frac{a^2 - b^2}{ab}

.

2

Set

f(k) = -\frac{7}{12}

and solve for the positive constant

k

.

\frac{-2k - 1}{k^2 + k} = -\frac{7}{12} \implies \frac{2k + 1}{k^2 + k} = \frac{7}{12} \implies 12(2k + 1) = 7(k^2 + k) \implies 7k^2 - 17k - 12 = 0

Cross-multiply and rearrange into a standard quadratic equation.

3

Factor the quadratic equation to find the positive root

k

.

(7k + 4)(k - 3) = 0 \implies k = -\frac{4}{7}

or

k = 3

. Since

k > 0

,

k = 3

.

The question specifies that

k

is a positive constant.

4

Evaluate

(k + 2) \Delta k

for

k = 3

.

(3 + 2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \times 3} = \frac{25 - 9}{15} = \frac{16}{15}

Substitute

a = 5

and

b = 3

into the custom operator definition.

Key Concept

Function definition substitution, non-commutative custom operators, and quadratic solving.

Estimated Time:2m 30s

Answer

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