All practice questions

Question 241Question →

If $n$ is a positive integer with the prime factorization $n = 2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers, and $n$ has exactly $40$ positive integer divisors, what is the minimum possible value of $a + b + c$ ?

7

8

9

11

14

Show answer & explanation

Answer: 8

Answer

8

The number of positive integer divisors of a number with prime factorization

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

is given by

(a+1)(b+1)(c+1)

. Setting this product equal to 40 and factoring 40 into three integers greater than 1 yields two distinct sets of factors:

\{2, 2, 10\}

and

\{2, 4, 5\}

. The sum of the factors in

\{2, 4, 5\}

is

2 + 4 + 5 = 11

, which is smaller than

2 + 2 + 10 = 14

. Subtracting 1 from each factor gives the exponents

a, b, c

as

1, 3, 4

, making the minimum sum

a+b+c = 1 + 3 + 4 = 8

.

Step-by-Step Solution

1

Express the number of divisors of

n

in terms of its exponents.

For

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

, the total number of positive divisors is

(a+1)(b+1)(c+1) = 40

.

The total divisor count formula requires taking the product of each prime factor's exponent increased by 1.

2

Find all combinations of three integers greater than 1 whose product is 40.

Since

a, b, c \ge 1

, we have

a+1, b+1, c+1 \ge 2

. The integer factorizations of 40 into 3 factors

\ge 2

are

(2, 2, 10)

and

(2, 4, 5)

.

We must break down 40 into 3 integer components corresponding to

(a+1)

,

(b+1)

, and

(c+1)

.

3

Calculate the sum

(a+1)+(b+1)+(c+1)

for each combination to find the minimum sum.

For

(2, 2, 10)

, the sum is

2 + 2 + 10 = 14

. For

(2, 4, 5)

, the sum is

2 + 4 + 5 = 11

. The minimum sum of factors is 11.

Minimizing

(a+1)+(b+1)+(c+1)

directly minimizes

a+b+c

.

4

Subtract 3 to obtain the minimum possible value of

a+b+c

.

a+b+c = 11 - 3 = 8

.

Since

(a+1)+(b+1)+(c+1) = (a+b+c) + 3 = 11

, subtracting 3 yields

a+b+c = 8

.

Key Concept

Divisor Count Formula from Prime Factorization

Question 242Question →

For any positive integer $n$ , let $u(n)$ denote the units digit of the sum $7^n + 3^{n+1} + 2^{n+2}$ . What is the remainder when the sum $u(1) + u(2) + u(3) + \dots + u(100)$ is divided by 7?

6

5

2

0

1

Show answer & explanation

Answer: 6

Answer

The remainder when the sum is divided by 7 is 6.

Evaluating the units digit

u(n)

for

n = 1, 2, 3, 4

gives

u(1) = 4

,

u(2) = 2

,

u(3) = 6

, and

u(4) = 0

. This pattern of 4 values repeats indefinitely. The sum of one full cycle of 4 terms is

4 + 2 + 6 + 0 = 12

. Since there are 100 terms in total, there are

100 / 4 = 25

full cycles. The total sum is

25 \times 12 = 300

. Dividing 300 by 7 gives a quotient of 42 with a remainder of 6. Thus, the correct choice is 6.

Step-by-Step Solution

1

Determine the units digit cyclicity for each power term.

The units digits of

7^n

follow the 4-term cycle [7, 9, 3, 1]. The units digits of

3^{n+1}

follow [9, 7, 1, 3]. The units digits of

2^{n+2}

follow [8, 4, 2, 6].

Units digits of positive integer powers repeat periodically with a cycle length of 4.

2

Calculate

u(n)

for the first 4 terms to identify the repeating pattern of

u(n)

.

For

n=1

:

7+9+8 = 24 \implies u(1)=4

. For

n=2

:

9+7+4 = 20 \implies u(2)=0

. For

n=3

:

3+1+2 = 6 \implies u(3)=6

. For

n=4

:

1+3+6 = 10 \implies u(4)=0

. The sequence of

u(n)

is

[4, 0, 6, 0]

.

Evaluating individual terms determines the fundamental period and sum per period of

u(n)

.

3

Sum the values of

u(n)

over one complete cycle of 4 terms.

Sum of one cycle =

4 + 0 + 6 + 0 = 10

.

Finding the sum of a single period simplifies finding the total sum over 100 terms.

4

Calculate the total sum for 100 terms and determine its remainder modulo 7.

Since 100 terms contain

100 / 4 = 25

complete cycles, the total sum

S = 25 \times 10 = 250

. Dividing 250 by 7 gives

250 = 7 \times 35 + 5

? Wait:

u(2) = (9+7+4)

units digit is 0. Let's verify:

7^2=49

(9),

3^3=27

(7),

2^4=16

(6).

9+7+6=22 \implies u(2)=2

. Let's re-verify:

u(1)=4, u(2)=2, u(3)=6, u(4)=0

. Sum per cycle =

4+2+6+0 = 12

. Total sum

S = 25 \times 12 = 300

.

300 = 7 \times 42 + 6

. Remainder is 6.

Dividing the total sum of 300 by 7 yields a quotient of 42 and a remainder of 6.

Key Concept

Units Digit Cyclicity and Modular Arithmetic Sums

Estimated Time:1m 30s

Question 243Question →

If $x$ and $y$ are positive integers such that $2^x - 2^{x-2} = 3^y$ , what is the value of $2^{x+y} + 3^{x-y}$ ?

11

13

25

35

73

Show answer & explanation

Answer: 11

Answer

The value of

2^{x+y} + 3^{x-y}

is 11.

Factoring the left side of the equation yields

2^{x-2}(2^2 - 1) = 3 \cdot 2^{x-2}

. Setting this equal to

3^y

and dividing both sides by 3 gives

2^{x-2} = 3^{y-1}

. Because 2 and 3 are coprime bases, equality is possible only when both exponents equal zero. This determines

x = 2

and

y = 1

, both of which are positive integers. Substituting these values into

2^{x+y} + 3^{x-y}

results in

2^3 + 3^1 = 8 + 3 = 11

.

Step-by-Step Solution

1

Factor out the common term on the left side of the given equation

2^x - 2^{x-2} = 2^{x-2}(2^2 - 1) = 2^{x-2}(4 - 1) = 3 \cdot 2^{x-2}

Factoring out the smallest power of 2 simplifies the subtraction into a multiplication.

2

Equate the factored expression to the right side and simplify

3 \cdot 2^{x-2} = 3^y \implies 2^{x-2} = \frac{3^y}{3} = 3^{y-1}

Dividing both sides by 3 isolates the prime bases 2 and 3 on opposite sides of the equation.

3

Apply fundamental properties of prime factorizations

x - 2 = 0 \implies x = 2

and

y - 1 = 0 \implies y = 1

Since 2 and 3 are distinct prime numbers,

2^a = 3^b

can hold for integer exponents only if

a = 0

and

b = 0

.

4

Evaluate the target expression using

x = 2

and

y = 1

2^{2+1} + 3^{2-1} = 2^3 + 3^1 = 8 + 3 = 11

Substituting the unique positive integer solution pair

(2, 1)

into the expression yields the final value.

Key Concept

Uniqueness of Prime Base Exponent Equality

Question 244Question →

When a positive integer $n$ is divided by $4$ , the remainder is $3$ . What is the units digit of the expression $M = 3^{n+1} + 8^{2n+1} - 7^{n+2}$ ?

2

4

6

8

0

Show answer & explanation

Answer: 6

Answer

The units digit of the expression is 6.

Since

n \equiv 3 \pmod{4}

, we substitute into each exponent:

n+1 \equiv 0 \pmod{4}

,

2n+1 \equiv 3 \pmod{4}

, and

n+2 \equiv 1 \pmod{4}

. The units digit cyclicity patterns give

3^4 \to 1

,

8^3 \to 2

, and

7^1 \to 7

. Evaluating the combined units digits yields

1 + 2 - 7 = -4

. Adding 10 to obtain a valid units digit gives

6

.

Step-by-Step Solution

1

Express the integer

n

in terms of modular arithmetic

n \equiv 3 \pmod{4}

, which means

n = 4k + 3

for some non-negative integer

k

.

Units digits of powers repeat in cycles of 4, so exponent behavior depends on exponents modulo 4.

2

Determine the units digit of

3^{n+1}

Exponent

n+1 = (4k+3)+1 = 4k+4 \equiv 0 \pmod{4}

. The units digit pattern for 3 is

3, 9, 7, 1

. Since exponent mod 4 is 0 (or 4), the units digit is

1

.

The cyclicity pattern for powers of 3 has period length 4.

3

Determine the units digit of

8^{2n+1}

Exponent

2n+1 = 2(4k+3)+1 = 8k+7 \equiv 3 \pmod{4}

. The units digit pattern for 8 is

8, 4, 2, 6

. For exponent mod 4 equal to 3, the units digit is

2

.

The cyclicity pattern for powers of 8 has period length 4.

4

Determine the units digit of

7^{n+2}

Exponent

n+2 = (4k+3)+2 = 4k+5 \equiv 1 \pmod{4}

. The units digit pattern for 7 is

7, 9, 3, 1

. For exponent mod 4 equal to 1, the units digit is

7

.

The cyclicity pattern for powers of 7 has period length 4.

5

Combine the units digits and resolve negative intermediate values

1 + 2 - 7 = -4 \equiv 6 \pmod{10}

. The units digit is

6

.

Units digits must be non-negative integers from 0 to 9, so a negative result requires adding 10.

Key Concept

Units Digit Cyclicity and Modular Arithmetic

Estimated Time:2m 0s

Question 245Question →

If $x$ and $y$ are positive integers such that $3^x \cdot 5^y = 675$ , what is the value of $2^{x+y} \cdot 3^{x-y}$ ?

32

36

96

192

288

Show answer & explanation

Answer: 96

Answer

96

Prime factorizing

675

gives

3^3 \cdot 5^2

. Equating the exponents yields

x = 3

and

y = 2

. Substituting these into

2^{x+y} \cdot 3^{x-y}

yields

2^5 \cdot 3^1 = 32 \cdot 3 = 96

.

Step-by-Step Solution

1

Find the prime factorization of 675.

675 = 27 \times 25 = 3^3 \cdot 5^2

Decomposing 675 into prime factors allows matching the exponents of prime bases 3 and 5.

2

Equate exponents of matching prime bases to determine

x

and

y

.

x = 3

and

y = 2

Since 3 and 5 are prime numbers, the prime factorization of 675 is unique.

3

Evaluate the target expression

2^{x+y} \cdot 3^{x-y}

.

2^{3+2} \cdot 3^{3-2} = 2^5 \cdot 3^1 = 32 \cdot 3 = 96

Substitute the determined values of

x

and

y

into the target power expression.

Key Concept

Prime Factorization and Exponent Properties

Estimated Time:1m 30s

Question 246Question →

If $n = 3^x \cdot 5^y \cdot 7^z$ , where $x$ , $y$ , and $z$ are positive integers, and $n$ has exactly $24$ positive divisors, what is the minimum possible value of $x + y + z$ ?

5

6

7

9

21

Show answer & explanation

Answer: 6

Answer

6

The total number of positive divisors of

n = 3^x \cdot 5^y \cdot 7^z

is

(x+1)(y+1)(z+1) = 24

. Because

x, y, z

are positive integers, each factor term must be at least 2. Factoring 24 into three integer factors each

\ge 2

yields two possibilities:

(2, 2, 6)

and

(2, 3, 4)

. The sum of the exponents

x+y+z

equals

(x+1)+(y+1)+(z+1) - 3

. For

(2, 2, 6)

, the sum is

2+2+6-3 = 7

. For

(2, 3, 4)

, the sum is

2+3+4-3 = 6

. The minimum possible value is 6.

Step-by-Step Solution

1

Express the number of divisors using the prime factorization formula.

For

n = 3^x \cdot 5^y \cdot 7^z

, the number of positive divisors is

(x+1)(y+1)(z+1) = 24

.

The total number of divisors of a prime-factored integer is found by taking the product of each prime factor's exponent plus one.

2

Determine the constraints on the factor terms.

Since

x, y, z

are positive integers (

x, y, z \ge 1

), each term

(x+1), (y+1), (z+1) \ge 2

.

Exponents must be at least 1, so each factor in the product must be at least 2.

3

Find all valid sets of 3 integer factors of 24 that are all

\ge 2

.

The possible triples

(a, b, c)

such that

a \cdot b \cdot c = 24

and

a, b, c \ge 2

are

(2, 2, 6)

and

(2, 3, 4)

.

We factor 24 into three integer components, each at least 2.

4

Calculate

x+y+z

for each factor triple to find the minimum.

For

(2, 2, 6)

:

x+1=2, y+1=2, z+1=6 \implies x+y+z = 1 + 1 + 5 = 7

.
For

(2, 3, 4)

:

x+1=2, y+1=3, z+1=4 \implies x+y+z = 1 + 2 + 3 = 6

.
The minimum value is 6.

Comparing the sums of exponents shows that the factor triple

(2, 3, 4)

minimizes

x+y+z

.

Key Concept

Divisors from Prime Factorization

Estimated Time:2m 0s

Question 247Question →

How many integer values of $x$ satisfy the inequality $\frac{|x - 2| - 3}{|x + 1| - 4} \leq 0$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

Analyzing the signs of the numerator

|x - 2| - 3

and denominator

|x + 1| - 4

shows that the quotient is non-positive when the numerator and denominator have opposite signs or when the numerator is zero. This yields two intervals:

(-5, -1]

and

(3, 5]

. The integers contained in these intervals are

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

, which totals 6 integer values.

Step-by-Step Solution

1

Analyze the sign of the numerator

|x - 2| - 3

The numerator is zero at

x = -1

and

x = 5

, negative for

-1 < x < 5

, and positive for

x < -1

or

x > 5

.

Solving

|x - 2| = 3

yields

x - 2 = 3 \implies x = 5

and

x - 2 = -3 \implies x = -1

.

2

Analyze the sign of the denominator

|x + 1| - 4

The denominator is zero at

x = -5

and

x = 3

, negative for

-5 < x < 3

, and positive for

x < -5

or

x > 3

. Exclude

x = -5

and

x = 3

.

Solving

|x + 1| = 4

yields

x + 1 = 4 \implies x = 3

and

x + 1 = -4 \implies x = -5

. Denominators cannot be zero.

3

Find intervals where numerator and denominator have opposite signs or numerator is zero

The solution set is the union of

(-5, -1]

and

(3, 5]

.

A fraction

\frac{N}{D} \leq 0

requires

N \ge 0, D < 0

or

N \le 0, D > 0

.

4

Count the total number of integer solutions

The valid integers are

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

, giving a total count of 6.

Listing integers in

(-5, -1]

yields

-4, -3, -2, -1

, and in

(3, 5]

yields

4, 5

.

Key Concept

Solving Rational Inequalities with Absolute Values

Question 248Question →

Consider the system of linear equations in

x

and

y

:

kx + 6y = 18

(k + 1)x + 9y = 27

where

k

is a constant. If this system has infinitely many solutions

(x, y)

, what is the sum of all real values of

x

that satisfy the equation

|kx - 10| = 3x - 4

?

\frac{14}{5}

-\frac{16}{5}

-6

10

\frac{6}{5}

Show answer & explanation

Answer: $\frac{14}{5}$

Answer

The sum of all real values of

x

satisfying the equation is

\frac{14}{5}

(or

2.8

).

The system of linear equations has infinitely many solutions when the ratio of coefficients is constant:

\frac{k}{k+1} = \frac{6}{9} = \frac{18}{27} = \frac{2}{3}

, which yields

k = 2

. Substituting

k = 2

gives

|2x - 10| = 3x - 4

. Breaking this into cases yields

2x - 10 = 3x - 4 \implies x = -6

and

-(2x - 10) = 3x - 4 \implies x = \frac{14}{5}

. Testing

x = -6

in the original equation gives

|-22| = -22

, which is false, making

x = -6

an extraneous solution. Testing

x = \frac{14}{5}

gives

|-4.4| = 4.4

, which is true. Thus, the only real solution is

\frac{14}{5}

, making the option equal to

\frac{14}{5}

correct.

Step-by-Step Solution

1

Determine the value of the constant

k

from the system of linear equations.

k = 2

For a system of two linear equations in two variables to have infinitely many solutions, the ratio of coefficients of

x

,

y

, and the constant terms must all be equal:

\frac{k}{k+1} = \frac{6}{9} = \frac{18}{27}

. Since

\frac{6}{9} = \frac{2}{3}

, setting

\frac{k}{k+1} = \frac{2}{3}

gives

3k = 2k + 2 \implies k = 2

.

2

Substitute

k = 2

into the absolute value equation.

|2x - 10| = 3x - 4

Replacing

k

with

2

sets up the single-variable linear absolute value equation to solve.

3

Solve Case 1 where the expression inside the absolute value is non-negative (

2x - 10 \ge 0

).

x = -6

(Extraneous)

Setting

2x - 10 = 3x - 4

yields

x = -6

. However, substituting

x = -6

back into the right side gives

3(-6) - 4 = -22

. Since an absolute value cannot equal a negative number,

x = -6

is an extraneous solution.

4

Solve Case 2 where the expression inside the absolute value is negative (

2x - 10 < 0

).

x = \frac{14}{5}

(Valid)

Setting

-(2x - 10) = 3x - 4

gives

-2x + 10 = 3x - 4 \implies 5x = 14 \implies x = \frac{14}{5}

. Substituting

x = 2.8

into the right side yields

3(2.8) - 4 = 4.4 \ge 0

, and

|2(2.8) - 10| = |-4.4| = 4.4

. Thus,

x = \frac{14}{5}

is valid.

5

Sum all valid real solutions.

\frac{14}{5}

Since

x = \frac{14}{5}

is the only valid solution, the sum of all solutions is

\frac{14}{5}

.

Key Concept

Systems of linear equations with infinitely many solutions require identical proportional coefficients. Absolute value equations of the form

|A| = B

require checking candidate solutions in

B \ge 0

to eliminate extraneous roots.

Estimated Time:3m 0s

Question 249Question →

When the integer $N = 4^{43} + 9^{43}$ is divided by $7$ , what is the remainder?

Show answer & explanation

Answer: 6

Answer

The remainder when

N = 4^{43} + 9^{43}

is divided by

7

is

6

.

Reducing

9 \pmod 7

yields

2

, allowing

N

to be expressed as

2^{86} + 2^{43} \pmod 7

. The powers of

2 \pmod 7

repeat every 3 powers

(2, 4, 1)

. Reducing the exponents

86

and

43

modulo 3 gives remainders of

2

and

1

, corresponding to values of

4

and

2

. Their sum

4 + 2 = 6

is the final remainder.

Step-by-Step Solution

1

Simplify the base modulo 7 and express in powers of 2

9 \equiv 2 \pmod 7

, so

N = 4^{43} + 9^{43} \equiv 2^{86} + 2^{43} \pmod 7

.

Reducing bases modulo 7 simplifies calculating large exponents.

2

Determine the remainder cyclicity of powers of 2 modulo 7

The cycle length is 3 with pattern

(2, 4, 1)

, because

2^1 \equiv 2

,

2^2 \equiv 4

, and

2^3 \equiv 1 \pmod 7

.

Powers of integers modulo a divisor repeat periodically.

3

Evaluate each term using the exponent modulo the cycle length

86 \equiv 2 \pmod 3 \implies 2^{86} \equiv 2^2 = 4 \pmod 7

, and

43 \equiv 1 \pmod 3 \implies 2^{43} \equiv 2^1 = 2 \pmod 7

.

The position in the cyclicity sequence is dictated by the exponent modulo the period length.

4

Add the individual remainders

4 + 2 = 6 \pmod 7

.

The remainder of a sum equals the sum of the individual remainders.

Key Concept

Modular arithmetic cyclicity of powers and addition of remainders

Question 250Question →

A technology manufacturing company produces two server models: Model Alpha and Model Beta. The manufacturing cost of one Model Alpha server is $400 more than twice the manufacturing cost of one Model Beta server. Last month, the company produced 15 Model Alpha servers and 40 Model Beta servers, incurring a total manufacturing cost of$ 76,000. What is the manufacturing cost, in dollars, of one Model Alpha server?

Show answer & explanation

Answer: 2400

Answer

The manufacturing cost of one Model Alpha server is $2,400.

By defining the costs as

A = 2B + 400

and setting up the total cost equation

15A + 40B = 76,000

, substituting

A

yields

70B = 70,000

, giving

B = 1,000

. Substituting

B = 1,000

back into

A = 2B + 400

gives

A = 2,400

.

Step-by-Step Solution

1

Define variables and construct the linear relation equation.

A = 2B + 400

The cost of one Model Alpha server (

A

) is

400 more than twice the cost of Model Beta (

B$).

2

Formulate the total production cost equation.

15A + 40B = 76,000

15 units of Model Alpha and 40 units of Model Beta sum to a total cost of $76,000.

3

Substitute

A

into the total cost equation and solve for

B

.

15(2B + 400) + 40B = 76,000 \implies 70B + 6,000 = 76,000 \implies B = 1,000

Substituting reduces the system to a single linear equation in one variable.

4

Calculate the value of

A

.

A = 2(1,000) + 400 = 2,400

Substitute

B = 1,000

back into the equation for

A

.

Key Concept

Solving Systems of Linear Equations by Substitution

Question 251Question →

If $4^x + 4^{x+1} + 4^{x+2} = 84 \cdot 2^{10}$ , what is the value of $x$ ?

5

6

7

10

12

Show answer & explanation

Answer: 6

Answer

6

Factoring out

4^x

gives

4^x(1 + 4 + 16) = 21 \cdot 4^x

. Dividing both sides of

21 \cdot 4^x = 84 \cdot 2^{10}

by 21 yields

4^x = 4 \cdot 2^{10}

. Converting terms to base 2 produces

2^{2x} = 2^2 \cdot 2^{10} = 2^{12}

. Setting exponents equal gives

2x = 12

, which yields

x = 6

.

Step-by-Step Solution

1

Factor out the common term

4^x

from the left side of the equation.

4^x(1 + 4^1 + 4^2) = 4^x(1 + 4 + 16) = 21 \cdot 4^x

Expressions adding powers with consecutive exponents can be simplified by factoring out the lowest power.

2

Substitute the factored expression into the equation and solve for

4^x

.

21 \cdot 4^x = 84 \cdot 2^{10} \implies 4^x = \frac{84}{21} \cdot 2^{10} = 4 \cdot 2^{10}

Dividing both sides by 21 isolates the exponential term on the left.

3

Convert both sides of the equation to powers of base 2.

(2^2)^x = 2^2 \cdot 2^{10} \implies 2^{2x} = 2^{12}

Expressing both sides with a common base enables equating exponents.

4

Equate exponents and solve for

x

.

2x = 12 \implies x = 6

When bases are equal, the powers must be equal.

Key Concept

Factoring Exponential Expressions and Base Conversion

Estimated Time:1m 30s

Question 252Question →

If $x$ , $y$ , and $z$ are integers satisfying $x < y < 0 < z$ , $x + y + z = 0$ , and $x y z = 160$ , what is the value of $z - x$ ?

Show answer & explanation

Answer: 18

Answer

The value of

z - x

is

18

.

By defining positive variables

a = -x

and

b = -y

, the given inequality

x < y < 0

implies

a > b > 0

. Since

x + y + z = 0

,

z = a + b

. The product condition

xyz = 160

translates to

ab(a+b) = 160

. Testing positive integer values reveals that

b = 2

and

a = 8

is the unique solution satisfying

a > b > 0

. This gives

x = -8

,

y = -2

, and

z = 10

, making

z - x = 10 - (-8) = 18

.

Step-by-Step Solution

1

Set up positive variables for the negative integers

Let

a = -x

and

b = -y

, where

a

and

b

are positive integers with

a > b > 0

. From

x + y + z = 0

, we get

-a - b + z = 0

, so

z = a + b

.

Converting negative integers to positive magnitude variables simplifies sign analysis in products and sums.

2

Substitute variables into the product equation

Substituting

x = -a

,

y = -b

, and

z = a + b

into

x y z = 160

gives

(-a)(-b)(a + b) = 160

, which simplifies to

a b (a + b) = 160

.

Multiplying two negative numbers yields a positive product, simplifying the product expression.

3

Solve for positive integer pairs

(a, b)

with

a > b

Testing integer values of

b

:
- If

b = 1

,

a(a+1) = 160

(no integer solution as

12 \times 13 = 156

).
- If

b = 2

,

2a(a+2) = 160 \implies a(a+2) = 80 \implies a = 8

.
- If

b = 3

,

3a(a+3) = 160 \implies a(a+3) = 53.33

(not an integer).
- If

b = 4

,

4a(a+4) = 160 \implies a(a+4) = 40

(no integer solution).
- If

b \ge 5

,

a > b \implies a \ge 6

, so

a b (a+b) \ge 6 \times 5 \times 11 = 330 > 160

.
Thus, the only valid integer pair is

a = 8

and

b = 2

.

Systematically checking integer factors under inequality constraints guarantees finding all unique solutions.

4

Calculate the target expression

z - x

Since

a = 8

and

b = 2

, we have

x = -8

,

y = -2

, and

z = 8 + 2 = 10

. Therefore,

z - x = 10 - (-8) = 18

.

Evaluating

z - x

using the identified values completes the solution.

Key Concept

Positive and Negative Number Properties and Inequality Constraints

Question 253Question →

If $k$ is a positive integer that is divisible by both 8 and 9, and $k$ has exactly 12 positive integer divisors, what is the value of $k$ ?

36

72

108

144

216

Show answer & explanation

Answer: 72

Answer

72

The value 72 has the prime factorization

2^3 \cdot 3^2

. It is divisible by

8 = 2^3

and

9 = 3^2

, and its number of positive integer divisors is

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

, satisfying all conditions of the problem.

Step-by-Step Solution

1

Express the divisibility conditions in terms of prime factors.

Since

k

is divisible by

8 = 2^3

and

9 = 3^2

, the prime factorization of

k

must be of the form

k = 2^a \cdot 3^b \cdot p_1^{c_1} \cdots

, where

a \geq 3

and

b \geq 2

.

Any multiple of 8 must contain at least three factors of 2, and any multiple of 9 must contain at least two factors of 3.

2

Write the formula for the total number of positive integer divisors.

The number of positive integer divisors is given by

(a+1)(b+1)(c_1+1)\cdots = 12

.

The total number of divisors of a number expressed in prime factorization is found by adding 1 to each exponent and multiplying the results.

3

Determine the values of the exponents

a

and

b

.

Since

a \geq 3

, we have

a+1 \geq 4

. Since

b \geq 2

, we have

b+1 \geq 3

. Therefore,

(a+1)(b+1) \geq 4 \times 3 = 12

.

Because the product

(a+1)(b+1)

is already at least 12, there can be no additional prime factors, so

a+1 = 4 \implies a = 3

and

b+1 = 3 \implies b = 2

.

4

Calculate the value of

k

.

k = 2^3 \cdot 3^2 = 8 \cdot 9 = 72

.

Evaluating

2^3 \cdot 3^2

gives the unique positive integer satisfying all given conditions.

Key Concept

Determining integer values using prime factorization constraints and total divisor count rules.

Estimated Time:2m 0s

Question 254Question →

Set $S$ consists of $n$ consecutive integers. The sum of all elements in set $S$ is equal to $0$ . If the product of the smallest element in set $S$ and the total number of elements $n$ is $-300$ , how many positive integers are in set $S$ ?

11

13

12

24

25

Show answer & explanation

Answer: 12

Answer

12

For a set of consecutive integers to sum to zero, the elements must be symmetric about zero. Therefore, the set has an odd number of terms

n = 2k + 1

ranging from

-k

to

k

. Setting up the product of the smallest term

(-k)

and the term count

(2k + 1)

gives

(-k)(2k + 1) = -300

, which simplifies to

2k^2 + k - 300 = 0

. Factoring this quadratic yields

(2k + 25)(k - 12) = 0

, so

k = 12

. The positive integers in the set are

1

through

12

, totaling 12 elements.

Step-by-Step Solution

1

Determine the symmetry and structure of set

S

using its sum.

The median of set

S

is

0

, and the set contains an odd number of terms

n = 2k + 1

centered at

0

, expressed as

\{-k, -(k-1), \dots, 0, \dots, k-1, k\}

.

For a set of consecutive integers to sum to

0

, the terms must be symmetric around

0

. Since

0

is an element of the set,

n

must be odd.

2

Set up the quadratic equation using the given product.

The smallest element is

-k

and the total number of elements is

n = 2k + 1

, yielding

(-k)(2k + 1) = -300

, which simplifies to

2k^2 + k - 300 = 0

.

The question specifies that the product of the smallest element and the number of elements is

-300

.

3

Solve the quadratic equation for

k

.

Factoring

(2k + 25)(k - 12) = 0

gives

k = 12

as the only positive integer solution.

Since

k

represents a count of elements strictly above zero,

k

must be a positive integer.

4

Count the number of positive integers in set

S

.

The positive integers are

1, 2, \dots, 12

, which gives a total of

12

positive integers.

Zero is neither positive nor negative, so only the integers from

1

to

k

(

12

) are counted.

Key Concept

Symmetry and median property of consecutive integers centered at zero.

Question 255Question →

An express train travels a distance of $180$ miles from Station A to Station B at a constant speed of $v$ miles per hour. On the return trip from Station B to Station A, the train travels along the same route at a constant speed that is $15$ miles per hour slower. If the return trip takes $1$ hour longer than the outbound trip, what is the value of $v$ ?

45

50

60

75

90

Show answer & explanation

Answer: $60$

Answer

60

The correct answer is

60

. The outbound time is

\frac{180}{v}

hours and the return time is

\frac{180}{v - 15}

hours. Setting their difference equal to

1

gives

\frac{180}{v - 15} - \frac{180}{v} = 1

. Solving the resulting quadratic equation

v^2 - 15v - 2700 = 0

yields

v = 60

miles per hour for the positive root.

Step-by-Step Solution

1

Express the time taken for each leg of the journey in terms of

v

.

Outbound time

t_1 = \frac{180}{v}

hours, return time

t_2 = \frac{180}{v - 15}

hours.

Time is equal to distance divided by speed.

2

Set up the equation based on the given time difference of

1

hour.

\frac{180}{v - 15} - \frac{180}{v} = 1

The return trip takes

1

hour longer than the outbound trip.

3

Clear the denominators by multiplying the equation by

v(v - 15)

.

180v - 180(v - 15) = v(v - 15) \implies 2700 = v^2 - 15v

Simplifying rational expressions into standard quadratic form.

4

Rearrange into

v^2 - 15v - 2700 = 0

and factor to solve for

v

.

(v - 60)(v + 45) = 0 \implies v = 60 \text{ or } v = -45

Factoring the quadratic equation.

5

Select the valid positive speed value.

Since speed must be positive,

v = 60

miles per hour.

Negative values for speed are not physically valid in this context.

Key Concept

Algebraic Word Problems and Equation Modeling

Question 256Question →

A logistics company ships three types of packages: Small, Medium, and Large. A shipment containing $3$ Small, $5$ Medium, and $2$ Large packages has a total weight of $170$ kilograms. A second shipment containing $1$ Small, $2$ Medium, and $1$ Large package has a total weight of $65$ kilograms. What is the total weight, in kilograms, of a shipment containing $5$ Small, $9$ Medium, and $4$ Large packages?

235

275

300

335

405

Show answer & explanation

Answer: $300$

Answer

The total weight of the shipment is

300

kilograms.

The target quantity

5S + 9M + 4L

can be expressed directly as

1 \times (3S + 5M + 2L) + 2 \times (1S + 2M + 1L)

. Substituting the given total weights yields

1(170) + 2(65) = 300

kilograms.

Step-by-Step Solution

1

Formulate linear equations representing the package weights.

Let

S

,

M

, and

L

represent the weight of a Small, Medium, and Large package respectively. Equation 1:

3S + 5M + 2L = 170

. Equation 2:

1S + 2M + 1L = 65

.

Setting up algebraic representations for the system based on the problem statement.

2

Determine if the requested expression

5S + 9M + 4L

can be formed as a linear combination

c_1(3S + 5M + 2L) + c_2(1S + 2M + 1L)

.

Match coefficients:

3c_1 + c_2 = 5

,

5c_1 + 2c_2 = 9

, and

2c_1 + c_2 = 4

.

Since individual values of

S

,

M

, and

L

cannot be uniquely determined from two equations with three variables, we seek scalar constants

c_1

and

c_2

.

3

Solve for the multipliers

c_1

and

c_2

.

Subtracting

2c_1 + c_2 = 4

from

3c_1 + c_2 = 5

yields

c_1 = 1

. Substituting

c_1 = 1

into

2c_1 + c_2 = 4

gives

c_2 = 2

. Checking the middle equation:

5(1) + 2(2) = 9

, which holds true.

Finding the scalar weights that recreate the exact combination requested.

4

Compute the total weight using the linear combination of the known values.

Total Weight

= 1 \times 170 + 2 \times 65 = 170 + 130 = 300

kilograms.

Applying the solved linear combination to the total weights of the shipments.

Key Concept

Solving for a linear combination of variables in an underdetermined system without finding individual variable values.

Question 257Question →

If $x$ is a real number that satisfies the equation $|x + 7| = 2x + 3$ , what is the sum of all valid real solutions for $x$ ?

\frac{2}{3}

4

-\frac{10}{3}

-\frac{2}{3}

\frac{22}{3}

Show answer & explanation

Answer: $4$

Answer

The sum of all valid real solutions is 4.

To solve

|x + 7| = 2x + 3

, we break the absolute value into two linear cases:

x + 7 = 2x + 3

giving

x = 4

, and

x + 7 = -(2x + 3)

giving

x = -\frac{10}{3}

. We must test both candidate solutions in the original equation because absolute value outputs must be non-negative. Testing

x = 4

gives

|11| = 11

, which is true. Testing

x = -\frac{10}{3}

gives

|\frac{11}{3}| = -\frac{11}{3}

, which is false because an absolute value cannot equal a negative number. Thus,

x = 4

is the only valid solution, making the sum of all valid solutions equal to

4

.

Step-by-Step Solution

1

Set up the two linear cases for the absolute value equation

|x + 7| = 2x + 3

.

Case 1:

x + 7 = 2x + 3

; Case 2:

x + 7 = -(2x + 3)

.

By definition,

|A| = B

implies

A = B

or

A = -B

, provided

B \ge 0

.

2

Solve Case 1 for

x

.

x + 7 = 2x + 3 \implies 2x - x = 7 - 3 \implies x = 4

.

Isolating

x

gives the first candidate solution.

3

Solve Case 2 for

x

.

x + 7 = -2x - 3 \implies 3x = -10 \implies x = -\frac{10}{3}

.

Expanding the negative sign and isolating

x

gives the second candidate solution.

4

Check candidate solutions in the original equation

|x + 7| = 2x + 3

to eliminate extraneous solutions.

For

x = 4

:

|4 + 7| = 11

and

2(4) + 3 = 11

(Valid).
For

x = -\frac{10}{3}

:

|-\frac{10}{3} + 7| = \frac{11}{3}

, but

2(-\frac{10}{3}) + 3 = -\frac{11}{3}

(Extraneous, since

\frac{11}{3} \neq -\frac{11}{3}

).

An absolute value expression cannot equal a negative number, so candidate values resulting in a negative right side are invalid.

5

Calculate the sum of all valid real solutions.

Sum =

4

.

Since

x = 4

is the only valid solution, the sum is simply

4

.

Key Concept

Solving Absolute Value Linear Equations and Validating against Extraneous Solutions

Estimated Time:2m 0s

Question 258Question →

If $x$ is a positive integer such that $\sqrt{\frac{8^{10} + 4^{10}}{8^4 + 4^{11}}} = 2^x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

Converting all terms to base 2 yields

\sqrt{\frac{2^{30}+2^{20}}{2^{12}+2^{22}}}

. Factoring out

2^{20}

in the numerator and

2^{12}

in the denominator leaves

\sqrt{\frac{2^{20}(2^{10}+1)}{2^{12}(2^{10}+1)}} = \sqrt{2^8} = 2^4

. Thus

2^x = 2^4

, so

x = 4

.

Step-by-Step Solution

1

Express all terms with prime base 2 using exponent rules.

8^{10} = 2^{30}

,

4^{10} = 2^{20}

,

8^4 = 2^{12}

, and

4^{11} = 2^{22}

.

Converting non-prime bases to a common base enables exponent simplification and factoring.

2

Factor the numerator and denominator by pulling out the lowest power of 2 in each.

Numerator:

2^{20}(2^{10} + 1)

; Denominator:

2^{12}(1 + 2^{10})

.

Factoring isolates common terms in sums of powers so they can be canceled.

3

Cancel the identical factor

(2^{10} + 1)

and compute the radical expression.

\sqrt{\frac{2^{20}}{2^{12}}} = \sqrt{2^8} = 2^4 = 16.

Applying quotient rule for exponents

\frac{2^{20}}{2^{12}} = 2^{8}

and radical rule

\sqrt{2^8} = (2^8)^{1/2} = 2^4

.

4

Set

2^4

equal to

2^x

to solve for

x

.

x = 4.

Since the bases are identical and positive, the exponents must be equal.

Key Concept

Exponents, Roots, and Powers of Integers

Question 259Question →

A positive integer $n$ is divisible by 12 and has exactly 15 positive integer divisors. What is the least possible value of $n$ ?

Show answer & explanation

Answer: 144

Answer

144

To find the least positive integer

n

that is a multiple of 12 and has 15 divisors, we examine the prime factorization of 12 (

2^2 \cdot 3^1

) and the divisor count formula

(e_1+1)(e_2+1)\dots = 15

. The number 15 factors as

5 \times 3

, implying

n

has the form

p^4 \cdot q^2

. Minimizing

n

requires placing the larger exponent on the smaller prime factor, so

p=2

and

q=3

, giving

n = 2^4 \cdot 3^2 = 144

.

Step-by-Step Solution

1

Determine prime factor requirements for divisibility by 12.

Since

12 = 2^2 \cdot 3^1

,

n

must have prime factors 2 and 3 with exponents

a \ge 2

and

b \ge 1

.

Divisibility requires that all prime factors of the divisor appear in the dividend with equal or higher exponents.

2

Apply the divisor count formula to determine the exponent structure.

The total number of divisors is 15, which factors as

(4 + 1)(2 + 1) = 15

. Thus,

n = p^4 \cdot q^2

.

The number of positive divisors is given by

(e_1 + 1)(e_2 + 1) \dots = 15

.

3

Assign prime factors to minimize

n

.

Assigning the larger exponent 4 to the smaller prime 2 gives

2^4 \cdot 3^2 = 144

.

To minimize a product of prime powers, larger exponents should be paired with smaller prime bases.

Key Concept

Divisor count formula and prime factorization properties

Question 260Question →

A hardware manufacturer produces two models of enterprise servers: Server X and Server Y. Each Server X requires $3$ processors of Type A and $2$ processors of Type B. Each Server Y requires $2$ processors of Type A and $5$ processors of Type B. In a given production run, the manufacturer assembled a total of $40$ servers using exactly $115$ processors of Type A. What is the total number of Type B processors used in this production run?

Show answer & explanation

Answer: 95

Answer

The total number of Type B processors used in the production run is 95.

By representing the count of Server X models as

x

and Server Y models as

y

, the given conditions yield two equations:

x + y = 40

and

3x + 2y = 115

. Solving this system gives

x = 35

and

y = 5

. Substituting these values into the expression for Type B processors (

2x + 5y

) produces

2(35) + 5(5) = 70 + 25 = 95

.

Step-by-Step Solution

1

Define variables and establish equations for server totals and Type A processor usage.

x + y = 40

and

3x + 2y = 115

, where

x

is Server X count and

y

is Server Y count.

Word problems require translating relationships into algebraic linear equations.

2

Solve the system of equations for

x

and

y

using substitution.

x = 35

Server X models and

y = 5

Server Y models.

Finding the specific quantities of each server model is required to compute total Type B processors.

3

Substitute

x = 35

and

y = 5

into the expression for Type B processors (

2x + 5y

).

2(35) + 5(5) = 70 + 25 = 95

.

Determines the target unknown value requested in the problem stem.

Key Concept

Modeling word problems using systems of linear equations in two variables.

Estimated Time:2m 0s

Answer

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