Arithmetic Practice Questions — GRE General Test

Which of the following rational numbers are strictly greater than $\frac{3}{5}$ ? Select all that apply.

Select all that apply

\frac{7}{10}

\frac{2}{3}

\frac{5}{9}

\frac{4}{7}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{7}{10}$ ; $\frac{2}{3}$ ; $\frac{11}{15}$

Answer

The fractions greater than 3/5 are 7/10, 2/3, and 11/15.

Converting 3/5 to decimal form yields 0.60. Evaluating the options: 7/10 equals 0.70, 2/3 is approximately 0.667, and 11/15 is approximately 0.733. Because each of these three values is strictly greater than 0.60, all three are correct choices.

Step-by-Step Solution

1

Convert the benchmark fraction 3/5 to decimal form.

3/5 = 0.60.

Decimal conversion provides a clear standard for comparing rational numbers.

2

Convert each given choice to decimal form and compare it to 0.60.

7/10 = 0.70; 2/3 ≈ 0.667; 5/9 ≈ 0.556; 4/7 ≈ 0.571; 11/15 ≈ 0.733.

Direct comparison reveals which decimals exceed 0.60.

3

Select all fractions whose values exceed 0.60.

The values 0.70, 0.667, and 0.733 are all strictly greater than 0.60.

The corresponding fractions 7/10, 2/3, and 11/15 satisfy the given condition.

Key Concept

Comparing Rational Numbers and Fractions

Question 2Question →

A baker has a flour mixture consisting only of wheat flour and rye flour. Currently, wheat flour accounts for $\frac{2}{5}$ of the total weight of the mixture. If the baker adds $9$ pounds of wheat flour to the mixture, wheat flour will account for $\frac{1}{2}$ of the new total weight of the mixture. What was the total weight, in pounds, of the original flour mixture?

Show answer & explanation

Answer: 45

Answer

The total weight of the original flour mixture was 45 pounds.

The initial total weight of the mixture is 45 pounds. Initially, wheat flour makes up

\frac{2}{5} \times 45 = 18

pounds. Adding 9 pounds of wheat flour increases the wheat flour to

18 + 9 = 27

pounds and the total weight to

45 + 9 = 54

pounds. The new fraction of wheat flour is

\frac{27}{54} = \frac{1}{2}

, which satisfies the given conditions.

Step-by-Step Solution

1

Express the initial weight of wheat flour in terms of the initial total weight

W

.

Initial weight of wheat flour =

\frac{2}{5}W

.

Wheat flour represents

\frac{2}{5}

of the total mixture.

2

Formulate an equation reflecting the addition of 9 pounds of wheat flour.

\frac{\frac{2}{5}W + 9}{W + 9} = \frac{1}{2}

Adding 9 pounds of wheat flour increases both the amount of wheat flour and the total weight of the mixture by 9 pounds.

3

Solve the algebraic equation for

W

.

Cross-multiplying gives

2\left(\frac{2}{5}W + 9\right) = W + 9

, which simplifies to

\frac{4}{5}W + 18 = W + 9

. Subtracting

\frac{4}{5}W

and

9

from both sides gives

\frac{1}{5}W = 9

, so

W = 45

.

Isolating

W

gives the value of the original total weight.

Key Concept

Setting up and solving equations involving fractional parts when a quantity is added to both the part and the whole.

Question 3Question →

How many positive integers $n$ less than $30$ satisfy the condition that $n^2 - 1$ is divisible by $24$ ?

5

8

9

10

15

Show answer & explanation

Answer: 10

Answer

10

For

n^2 - 1

to be divisible by

24 = 8 \times 3

,

n

must be an odd integer that is not divisible by 3, which means

\gcd(n, 6) = 1

. The positive integers less than 30 coprime to 6 are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29, totaling 10 integers.

Step-by-Step Solution

1

Factor the expression and analyze divisibility requirements for 24.

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

. Since

24 = 8 \times 3

with

\gcd(8, 3) = 1

,

n^2 - 1

must be divisible by both

8

and

3

.

Decompose 24 into coprime prime-power factors to evaluate modular conditions independently.

2

Determine the condition for

n^2 - 1

to be divisible by 8.

If

n

is even,

n^2 - 1

is odd and cannot be divisible by 8. If

n

is odd, let

n = 2k + 1

; then

n^2 - 1 = 4k(k + 1)

. Since one of

k

or

k + 1

is always even,

4k(k + 1)

is divisible by 8. Thus,

n

must be odd.

Analyze parity requirements for the power of 2.

3

Determine the condition for

n^2 - 1

to be divisible by 3.

If

n

is a multiple of 3,

n^2 - 1 \equiv -1 \pmod 3

, which is not divisible by 3. If

n

is not a multiple of 3,

n \equiv 1

or

2 \pmod 3

, so

n^2 \equiv 1 \pmod 3

and

n^2 - 1

is divisible by 3. Thus,

n

cannot be a multiple of 3.

Analyze remainder properties modulo 3.

4

Combine the conditions and count valid positive integers

n < 30

.

The combined requirement is

\gcd(n, 6) = 1

(

n

is odd and not a multiple of 3). The positive integers less than 30 satisfying this are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29. Counting these yields 10 integers.

Enumerate all integers meeting the coprime condition within the specified domain.

Key Concept

Modular arithmetic properties of quadratic expressions and coprimality constraints

Question 4Question →

Let $n$ be a positive integer whose prime factorization consists only of the prime factors $2$ and $3$ . If $n$ has exactly $12$ positive divisors and $\gcd(n, 36) = 12$ , what is the value of $n$ ?

Show answer & explanation

Answer: 96

Answer

The value of

n

is

96

.

Representing

n = 2^a \times 3^b

, the number of positive divisors is

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

. The greatest common divisor

\gcd(n, 36) = \gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 12 = 2^2 \times 3^1

. This requires

\min(a,2) = 2 \implies a \ge 2

and

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

. Substituting

b = 1

into

(a+1)(1+1) = 12

gives

2(a+1) = 12

, so

a = 5

. Therefore,

n = 2^5 \times 3^1 = 32 \times 3 = 96

.

Step-by-Step Solution

1

Set up the prime factorization of

n

and the divisor count equation.

n = 2^a \times 3^b

and

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

.

Since the prime factors of

n

are only

2

and

3

,

n

must take the form

2^a \times 3^b

, where the number of positive divisors is

(a+1)(b+1)

.

2

Analyze the exponent requirements using the greatest common divisor.

a \ge 2

and

b = 1

.

\gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 2^2 \times 3^1

. Matching powers gives

\min(a,2) = 2 \implies a \ge 2

, and

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

.

3

Solve for exponent

a

and calculate

n

.

a = 5

, giving

n = 2^5 \times 3^1 = 96

.

Substituting

b = 1

into

(a+1)(1+1) = 12

gives

2(a+1) = 12 \implies a = 5

, which satisfies

a \ge 2

.

Key Concept

Prime exponent rules for GCD and divisor counting

Estimated Time:2m 0s

Question 5Question →

Let $K = 2^5 \times 3^4 \times 5^3 \times 11^2$ . How many positive integer factors of $K$ are divisible by $300$ but are not divisible by $900$ ?

16

24

36

48

96

Show answer & explanation

Answer: 24

Answer

24

The correct answer 24 is obtained by analyzing the prime factorizations

300 = 2^2 \times 3^1 \times 5^2

and

900 = 2^2 \times 3^2 \times 5^2

. Any factor

f = 2^a \times 3^b \times 5^c \times 11^d

must satisfy

a \in \{2, 3, 4, 5\}

(4 choices),

b = 1

(1 choice, since

b \ge 1

for 300 but

b < 2

for 900),

c \in \{2, 3\}

(2 choices), and

d \in \{0, 1, 2\}

(3 choices). Multiplying these choices gives

4 \times 1 \times 2 \times 3 = 24

.

Step-by-Step Solution

1

Express any factor of

K

in terms of prime factor exponent constraints.

Any factor

f

of

K

has the form

f = 2^a \times 3^b \times 5^c \times 11^d

, where

0 \le a \le 5

,

0 \le b \le 4

,

0 \le c \le 3

, and

0 \le d \le 2

.

The prime factors of

f

must be subsets of the prime factors of

K

with exponents not exceeding those in

K

.

2

Find the prime factorizations of

300

and

900

.

300 = 2^2 \times 3^1 \times 5^2

and

900 = 2^2 \times 3^2 \times 5^2

.

Divisibility criteria correspond to minimum exponent requirements for each prime factor.

3

Apply the divisibility conditions to determine constraints on each exponent.

For

f

to be divisible by

300

, we need

a \ge 2

,

b \ge 1

, and

c \ge 2

. For

f

to NOT be divisible by

900

, we must have

b < 2

. Thus,

b = 1

.

Combining

b \ge 1

and

b < 2

uniquely restricts

b

to

1

.

4

Count the number of valid choices for each exponent.

a \in \{2, 3, 4, 5\}

(

4

choices),

b \in \{1\}

(

1

choice),

c \in \{2, 3\}

(

2

choices), and

d \in \{0, 1, 2\}

(

3

choices).

The exponent

d

is unrestricted by

300

or

900

, so it can take any valid power present in

K

.

5

Multiply the number of independent choices using the fundamental counting principle.

4 \times 1 \times 2 \times 3 = 24

.

Each exponent choice can be paired independently to form a unique factor.

Key Concept

Counting Divisors Using Prime Exponent Constraints

Question 6Question →

A beverage mixture is prepared by combining apple juice, orange juice, and grape juice in a ratio of $3 : 4 : 5$ by volume. If a pitcher contains a total of $48$ fluid ounces of this mixture, how many fluid ounces of orange juice are in the pitcher?

12

fluid ounces

16

fluid ounces

20

fluid ounces

24

fluid ounces

36

fluid ounces

Show answer & explanation

Answer: $16$ fluid ounces

Answer

16

fluid ounces

The ratio

3 : 4 : 5

indicates that the total mixture is divided into

3 + 4 + 5 = 12

equal parts. Orange juice accounts for

4

of these

12

parts, representing

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

of the total volume. Taking

\frac{1}{3}

of

48

fluid ounces gives

16

fluid ounces.

Step-by-Step Solution

1

Calculate the total number of ratio parts.

Total parts =

3 + 4 + 5 = 12

.

To find the fraction of each ingredient in the mixture, we must sum all parts of the ratio.

2

Determine the fraction of the total volume that represents orange juice.

Orange juice fraction =

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

.

Orange juice corresponds to

4

parts out of the

12

total parts.

3

Multiply the orange juice fraction by the total volume.

\frac{1}{3} \times 48 = 16

fluid ounces.

Multiplying the part-to-whole fraction by the total volume yields the exact volume of orange juice.

Key Concept

Part-to-Whole Ratio Relationships

Estimated Time:45s

Question 7Question →

If $x$ and $y$ are real numbers such that $|2x - 5| \le 9$ and $|y + 3| \le 4$ , what is the maximum possible value of $|x - 2y + 1|$ ?

22

20

14

13

6

Show answer & explanation

Answer: $22$

Answer

The maximum possible value of

|x - 2y + 1|

is 22.

The correct answer is 22 because solving the absolute value inequalities gives

-2 \le x \le 7

and

-7 \le y \le 1

. To maximize

x - 2y + 1

, we take the largest possible value of

x

(

7

), the smallest possible value of

y

(

-7

, making

-2y = 14

), and add

1

, obtaining

7 + 14 + 1 = 22

. Since the lower bound of

x - 2y + 1

is

-3

, the maximum absolute value is

|22| = 22

.

Step-by-Step Solution

1

Find the range of possible values for

x

from the inequality

|2x - 5| \le 9

.

-9 \le 2x - 5 \le 9 \implies -4 \le 2x \le 14 \implies -2 \le x \le 7

.

An absolute value inequality

|A| \le B

unwraps to

-B \le A \le B

.

2

Find the range of possible values for

y

from the inequality

|y + 3| \le 4

.

-4 \le y + 3 \le 4 \implies -7 \le y \le 1

.

Unwrapping the absolute value inequality gives the upper and lower bounds for

y

on the real number line.

3

Determine the range of possible values for

-2y

.

Multiplying

-7 \le y \le 1

by

-2

and reversing the inequality signs yields

-2(1) \le -2y \le -2(-7) \implies -2 \le -2y \le 14

.

Multiplying an inequality by a negative number reverses the direction of the inequality signs.

4

Combine the ranges to find the minimum and maximum bounds for

z = x - 2y + 1

.

Minimum

z = (-2) + (-2) + 1 = -3

; Maximum

z = 7 + 14 + 1 = 22

. Thus,

-3 \le x - 2y + 1 \le 22

.

Adding the individual minimums gives the absolute minimum, and adding the individual maximums gives the absolute maximum.

5

Calculate the maximum value of the absolute value

|x - 2y + 1|

.

\\max(|x - 2y + 1|) = \\max(|-3|, |22|) = 22

.

The absolute value of a quantity ranging from

-3

to

22

reaches its maximum distance from zero at

22

.

Key Concept

Properties of Real Numbers, Number Line Inequalities, and Absolute Value Operations

Estimated Time:2m 30s

Question 8Question →

A laboratory mixture is created by combining two solutions containing a specific tracer compound. Solution X has a volume of $2.5 \times 10^3$ milliliters and a tracer concentration of $4.8 \times 10^{-7}$ grams per milliliter. Solution Y has a volume of $7.5 \times 10^3$ milliliters and a tracer concentration of $1.6 \times 10^{-6}$ grams per milliliter. What is the total mass, in grams, of the tracer compound present in the combined mixture?

1.32 \times 10^{-2}

2.4 \times 10^{-3}

1.32 \times 10^{-5}

2.4 \times 10^{-2}

1.32 \times 10^{-1}

Show answer & explanation

Answer: $1.32 \times 10^{-2}$

Answer

1.32 \times 10^{-2}

grams

The mass in Solution X is

(2.5 \times 10^3) \times (4.8 \times 10^{-7}) = 1.2 \times 10^{-3}

grams, and the mass in Solution Y is

(7.5 \times 10^3) \times (1.6 \times 10^{-6}) = 1.2 \times 10^{-2}

grams. Rewriting

1.2 \times 10^{-3}

as

0.12 \times 10^{-2}

allows direct addition:

(0.12 + 1.2) \times 10^{-2} = 1.32 \times 10^{-2}

grams.

Step-by-Step Solution

1

Calculate the mass of the tracer in Solution X

M_X = (2.5 \times 10^3) \times (4.8 \times 10^{-7}) = 12.0 \times 10^{-4} = 1.2 \times 10^{-3}

grams

Mass equals volume multiplied by concentration.

2

Calculate the mass of the tracer in Solution Y

M_Y = (7.5 \times 10^3) \times (1.6 \times 10^{-6}) = 12.0 \times 10^{-3} = 1.2 \times 10^{-2}

grams

Mass equals volume multiplied by concentration.

3

Sum the tracer masses and adjust to scientific notation

M_{\text{total}} = 1.2 \times 10^{-3} + 1.2 \times 10^{-2} = 0.12 \times 10^{-2} + 1.2 \times 10^{-2} = 1.32 \times 10^{-2}

grams

To add terms in scientific notation, convert them to have matching exponents before adding coefficients.

Key Concept

Arithmetic operations with Scientific Notation and Decimals

Estimated Time:1m 30s

Question 9Question →

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}}$ for all positive integers $n$ . If $S = \sum_{n=1}^{98} a_n$ , which of the following is closest to the value of $S$ when rounded to the nearest tenth?

8.8

4.5

9.0

9.9

17.5

Show answer & explanation

Answer: 8.8

Answer

The value of

S

rounded to the nearest tenth is 8.8.

The term

a_n

simplifies to

\frac{\sqrt{n+2} - \sqrt{n}}{2}

upon rationalizing the denominator. Summing from

n=1

to

98

causes all middle terms to cancel out, leaving

\frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2}

. Substituting

\sqrt{100} = 10

,

\sqrt{1} = 1

,

\sqrt{99} \approx 9.95

, and

\sqrt{2} \approx 1.41

gives approximately

8.77

, which rounds to 8.8.

Step-by-Step Solution

1

Rationalize the general term

a_n

a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}} \cdot \frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} - \sqrt{n}} = \frac{\sqrt{n+2} - \sqrt{n}}{(n+2) - n} = \frac{\sqrt{n+2} - \sqrt{n}}{2}

Eliminating radicals from the denominator reveals the underlying telescoping structure of the sequence.

2

Expand the summation

S = \sum_{n=1}^{98} a_n

S = \frac{1}{2} \left[ (\sqrt{3} - \sqrt{1}) + (\sqrt{4} - \sqrt{2}) + (\sqrt{5} - \sqrt{3}) + \dots + (\sqrt{99} - \sqrt{97}) + (\sqrt{100} - \sqrt{98}) \right]

Writing out initial and final terms demonstrates which terms cancel.

3

Simplify the telescoping sum

S = \frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2} = \frac{\sqrt{99} + 10 - 1 - \sqrt{2}}{2} = \frac{9 + \sqrt{99} - \sqrt{2}}{2}

All intermediate terms cancel out, leaving two positive boundary terms and two negative boundary terms.

4

Estimate square root values and perform rounding

Since

\sqrt{99} \approx 9.94987

and

\sqrt{2} \approx 1.41421

,

S \approx \frac{9 + 9.94987 - 1.41421}{2} = \frac{17.53566}{2} = 8.76783 \approx 8.8

Evaluating the radicals to two decimal places allows accurate rounding to the nearest tenth.

Key Concept

Telescoping Series Summation and Square Root Estimation

Question 10Question →

If $x > 1$ and $\sqrt{x^{\sqrt{x}}} = x^2$ , what is the value of $x$ ?

2

4

8

16

64

Show answer & explanation

Answer: $16$

Answer

16

By rewriting the square root as an exponent of

1/2

, the left side becomes

x^{\frac{\sqrt{x}}{2}}

. Since the base

x

is greater than 1, set the exponents equal to each other:

\frac{\sqrt{x}}{2} = 2

, which gives

\sqrt{x} = 4

. Squaring both sides yields

x = 16

.

Step-by-Step Solution

1

Express the radical on the left side of the equation as a fractional exponent.

\sqrt{x^{\sqrt{x}}} = (x^{\sqrt{x}})^{\frac{1}{2}} = x^{\frac{\sqrt{x}}{2}}

The square root rule states that

\sqrt[k]{a} = a^{\frac{1}{k}}

.

2

Set the exponent of the left side equal to the exponent of the right side.

\frac{\sqrt{x}}{2} = 2

Since the bases are equal (

x > 1

), their corresponding exponents must be equal.

3

Solve for

\sqrt{x}

by multiplying both sides by 2.

\sqrt{x} = 4

Isolating the radical term allows determination of the root value.

4

Square both sides to solve for

x

.

x = 4^2 = 16

Squaring a principal square root yields the underlying radicand.

Key Concept

Combining Fractional Exponents and Radical Expressions

Question 11Question →

A renewable energy facility distributes stored electricity among three battery banks: Bank 1, Bank 2, and Bank 3. Initially, the ratio of the energy stored in Bank 1 to Bank 2 to Bank 3 is $3 : 4 : 5$ .

To balance the system, two sequential transfers of energy are performed without any energy loss:
1. Energy is transferred from Bank 3 to Bank 1 such that the ratio of the energy in Bank 1 to Bank 2 becomes $5 : 4$ .
2. Energy is then transferred from Bank 2 to Bank 3 such that the ratio of the energy in Bank 2 to Bank 3 becomes $1 : 4$ .

If Bank 3 contains $60\text{ MWh}$ more energy after the second transfer than it did initially, what was the total amount of energy, in MWh, stored across all three battery banks?

1,200\text{ MWh}

1,500\text{ MWh}

900\text{ MWh}

720\text{ MWh}

500\text{ MWh}

Show answer & explanation

Answer: $1,200\text{ MWh}$

Answer

The total energy stored across all three battery banks is

1,200\text{ MWh}

.

The correct answer is

1,200\text{ MWh}

. By setting up algebraic expressions for the energy in each bank relative to total energy

T

, we track how each transfer modifies individual bank totals while conserving total energy. The intermediate transfer reduces Bank 3 from

\frac{25}{60}T

to

\frac{15}{60}T

, and the subsequent transfer increases it to

\frac{28}{60}T

. The resulting difference of

\frac{3}{60}T = \frac{1}{20}T

equals

60\text{ MWh}

, solving directly to

T = 1,200\text{ MWh}

.

Step-by-Step Solution

1

Express initial energy quantities in terms of total energy

T

Bank 1 =

\frac{3}{12}T

, Bank 2 =

\frac{4}{12}T

, Bank 3 =

\frac{5}{12}T = \frac{25}{60}T

The ratio

3 : 4 : 5

sums to

12

total parts.

2

Calculate energy amounts after the first transfer (Bank 3 to Bank 1)

Bank 2 remains

\frac{4}{12}T

; Bank 1 becomes

\frac{5}{12}T

; Bank 3 becomes

T - \frac{5}{12}T - \frac{4}{12}T = \frac{3}{12}T

The new ratio of Bank 1 to Bank 2 is

5 : 4

, and Bank 2's energy did not change.

3

Calculate energy amounts after the second transfer (Bank 2 to Bank 3)

Combined energy in Banks 2 and 3 =

\frac{4}{12}T + \frac{3}{12}T = \frac{7}{12}T

. Final Bank 3 =

\frac{4}{5} \times \frac{7}{12}T = \frac{28}{60}T

Bank 1 remains unchanged at

\frac{5}{12}T

. The remaining energy is divided between Bank 2 and Bank 3 in a

1 : 4

ratio.

4

Determine the net change in Bank 3 and solve for total energy

T

Net change =

\frac{28}{60}T - \frac{25}{60}T = \frac{3}{60}T = \frac{1}{20}T

. Since

\frac{1}{20}T = 60\text{ MWh}

,

T = 1,200\text{ MWh}

Bank 3 ended with

60\text{ MWh}

more than its initial amount.

Key Concept

Multi-stage ratio rebalancing and conservation of total quantity

Question 12Question →

Let $p$ and $q$ be integers such that $p < 0 < q$ , $(-1)^p + (-1)^q = 0$ , and $p^2 q + p$ is an even integer. Which of the following expressions must be negative?

(-p)^q

(-q)^p

q^p

p^q

(q - p)^p

Show answer & explanation

Answer: $p^q$

Answer

p^q

From

(-1)^p + (-1)^q = 0

,

p

and

q

must have opposite parities. Factoring

p^2 q + p

into

p(pq + 1)

shows that if

p

were odd,

q

would be even, leading to an odd product

\text{odd} \times \text{odd} = \text{odd}

. Because

p^2 q + p

is given as even,

p

must be an even integer and

q

must be an odd integer. Given

p < 0

,

p

is a negative even integer, and

q

is a positive odd integer. The expression

p^q

represents a negative base raised to an odd exponent, which is guaranteed to be negative.

Step-by-Step Solution

1

Determine the parities of

p

and

q

using the equation

(-1)^p + (-1)^q = 0

.

One of

p

or

q

is even and the other is odd.

For the sum

(-1)^p + (-1)^q

to equal

0

, one term must equal

1

and the other must equal

-1

, meaning

p

and

q

have opposite parities.

2

Analyze the parity of

p^2 q + p = p(pq + 1)

to determine which variable is even.

p

must be even, and

q

must be odd.

If

p

were odd,

q

would be even, making

pq

even,

pq+1

odd, and

p(pq+1)

odd. Since

p^2 q + p

is even,

p

must be even and

q

must be odd.

3

Evaluate the sign of

p^q

using sign rules for exponents.

p^q < 0

p

is negative (

p < 0

) and

q

is a positive odd integer. A negative number raised to an odd power is always negative.

Key Concept

Parity and sign rules for negative bases and integer exponents

Question 13Question →

If $x$ is a real number such that $\sqrt[4]{16^{x+3}} = 8^{x-1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

To solve

\sqrt[4]{16^{x+3}} = 8^{x-1}

, express both sides with the base 2. The left side simplifies to

\sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

. The right side simplifies to

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

. Equating the exponents yields

x + 3 = 3x - 3

, which solves to

2x = 6

, giving

x = 3

.

Step-by-Step Solution

1

Rewrite 16 and 8 using prime base 2

16 = 2^4

and

8 = 2^3

Converting terms to a common base allows direct comparison of exponents.

2

Simplify the left-hand side radical expression

\sqrt[4]{16^{x+3}} = \sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

The

n

-th root

\sqrt[n]{a^m}

is equivalent to

a^{m/n}

.

3

Simplify the right-hand side exponential expression

8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x-3}

Applying the exponent power rule

(a^m)^n = a^{m \cdot n}

requires multiplying

3

by

(x - 1)

.

4

Equate the exponents and solve for

x

x + 3 = 3x - 3 \implies 2x = 6 \implies x = 3

When

a^u = a^v

for

a > 0

and

a \neq 1

, it follows that

u = v

.

Key Concept

Solving exponential equations using prime base factorization and radical conversion rules

Question 14Question →

How many positive integers less than $100$ are divisible by both $4$ and $6$ , but are NOT divisible by $8$ ?

Show answer & explanation

Answer: 4

Answer

The correct numerical answer is 4.

To be divisible by both 4 and 6, an integer must be a multiple of

\text{LCM}(4, 6) = 12

. The positive integers less than 100 that are multiples of 12 are 12, 24, 36, 48, 60, 72, 84, and 96 (8 integers). Among these, those divisible by 8 are multiples of

\text{LCM}(12, 8) = 24

, which are 24, 48, 72, and 96 (4 integers). Subtracting the excluded integers yields

8 - 4 = 4

.

Step-by-Step Solution

1

Find the least common multiple of 4 and 6.

LCM(4, 6) = 12

An integer divisible by both 4 and 6 must be a multiple of their least common multiple.

2

Count positive integers less than 100 that are multiples of 12.

The multiples are 12, 24, 36, 48, 60, 72, 84, and 96, giving 8 integers.

The largest multiple of 12 strictly less than 100 is 96 (12 × 8).

3

Identify multiples of 12 that are also divisible by 8.

Since LCM(12, 8) = 24, these are the multiples of 24: 24, 48, 72, and 96, giving 4 integers.

Any integer divisible by both 12 and 8 must be a multiple of 24.

4

Subtract the excluded integers from the total count.

8 - 4 = 4

We exclude the multiples of 8 from the set of multiples of 12.

Key Concept

Divisibility, Least Common Multiple (LCM), and Set Exclusion

Question 15Question →

What is the least positive integer $n$ that leaves a remainder of $3$ when divided by $7$ , a remainder of $4$ when divided by $5$ , and is divisible by $9$ ?

Show answer & explanation

Answer: 234

Answer

The least positive integer satisfying all three conditions is 234.

To find the least positive integer

n

that satisfies

n \equiv 3 \pmod{7}

,

n \equiv 4 \pmod{5}

, and

n \equiv 0 \pmod{9}

, we first find a general expression for integers meeting the first two conditions. Checking values of

5m + 4

modulo 7 gives

24

as the smallest positive integer matching both. The combined condition is

n \equiv 24 \pmod{35}

, or

n = 35k + 24

. Requiring

35k + 24

to be divisible by 9 gives

8k + 6 \equiv 0 \pmod{9}

, which simplifies to

k \equiv 6 \pmod{9}

. The smallest non-negative integer value for

k

is

6

, leading to

n = 35(6) + 24 = 234

.

Step-by-Step Solution

1

Set up system of modular congruences for the remainders

n \equiv 3 \pmod{7}

,

n \equiv 4 \pmod{5}

, and

n \equiv 0 \pmod{9}

Translates the remainder and divisibility conditions into mathematical equations.

2

Combine the first two congruences using the Chinese Remainder Theorem approach

n \equiv 24 \pmod{35}

, so

n = 35k + 24

for an integer

k \ge 0

Since

\text{lcm}(5, 7) = 35

, the solutions to the combined system repeat every 35 integer values.

3

Enforce the divisibility condition by 9 on

n = 35k + 24

35k + 24 \equiv 0 \pmod{9} \implies 8k + 6 \equiv 0 \pmod{9} \implies -k + 6 \equiv 0 \pmod{9} \implies k \equiv 6 \pmod{9}

Reduces coefficients modulo 9 to find the values of

k

that make

n

a multiple of 9.

4

Calculate the smallest positive integer

n

corresponding to

k = 6

n = 35(6) + 24 = 234

Choosing

k = 6

yields the smallest non-negative integer for

k

that satisfies all conditions.

Key Concept

Simultaneous congruences and divisibility constraints

Question 16Question →

If $n$ is an integer that is divisible by $15$ , which of the following statements MUST also be true? Select all such statements.

Select all that apply

n

is divisible by

3

n

is divisible by

5

n + 15

is divisible by

15

n

is divisible by

30

n

is an even integer

Show answer & explanation

Answer: $n$ is divisible by $3$ ; $n$ is divisible by $5$ ; $n + 15$ is divisible by $15$

Answer

The statements asserting that

n

is divisible by

3

,

n

is divisible by

5

, and

n + 15

is divisible by

15

must all be true.

Any integer divisible by

15

must be divisible by all factors of

15

, which includes

3

and

5

. Furthermore, adding

15

to a multiple of

15

yields another multiple of

15

, so

n + 15

is also divisible by

15

.

Step-by-Step Solution

1

Analyze the prime factor decomposition of the divisor

The prime factorization of

15

is

3 \times 5

. Therefore, any integer

n

that is a multiple of

15

can be expressed as

n = 15k = 3 \times 5 \times k

for some integer

k

.

Understanding factor relationships allows evaluation of divisibility rules.

2

Evaluate divisibility of

n

by

3

and

5

Since

n = 3(5k)

,

n

is divisible by

3

. Since

n = 5(3k)

,

n

is divisible by

5

. Both statements are always true.

Any multiple of a composite number is also a multiple of that number's factors.

3

Evaluate the expression

n + 15

n + 15 = 15k + 15 = 15(k + 1)

. Since

k + 1

is an integer,

n + 15

is a multiple of

15

. This statement is always true.

Adding a multiple of

15

to another multiple of

15

yields a multiple of

15

.

4

Test counterexamples for remaining statements

For

n = 15

:

15

is not divisible by

30

, so divisibility by

30

is not guaranteed. Additionally,

15

is odd, so being an even integer is not guaranteed.

A statement must hold for all possible values of

n

to be necessarily true.

Key Concept

Divisibility by a composite number implies divisibility by all of its factors, and multiples of a number remain multiples when another multiple of that number is added.

Question 17Question →

On the real number line, point $P$ represents the real number $x$ , point $Q$ represents $7$ , and point $R$ represents $-5$ . If the distance between $P$ and $Q$ is equal to $3$ times the distance between $P$ and $R$ , what is the sum of all possible values of $x$ ?

-13

-11

-7

-2

13

Show answer & explanation

Answer: $-13$

Answer

The sum of all possible values of

x

is

-13

.

The distance between

x

and

7

is

|x - 7|

and the distance between

x

and

-5

is

|x + 5|

. Equating

|x - 7| = 3|x + 5|

leads to two equations:

x - 7 = 3(x + 5)

giving

x = -11

, and

x - 7 = -3(x + 5)

giving

x = -2

. Adding both solutions yields

(-11) + (-2) = -13

.

Step-by-Step Solution

1

Set up the distance equation using absolute value notation

The distance between

P(x)

and

Q(7)

is

|x - 7|

, and the distance between

P(x)

and

R(-5)

is

|x - (-5)| = |x + 5|

. The problem specifies that

|x - 7| = 3|x + 5|

.

Distance between two points

a

and

b

on a real number line is expressed as

|a - b|

.

2

Solve Case 1 where

x - 7

and

x + 5

have the same sign

x - 7 = 3(x + 5) \implies x - 7 = 3x + 15 \implies -22 = 2x \implies x = -11

.

When both absolute value expressions have identical signs,

|x - 7| = 3|x + 5|

simplifies directly to

x - 7 = 3(x + 5)

.

3

Solve Case 2 where

x - 7

and

x + 5

have opposite signs

x - 7 = -3(x + 5) \implies x - 7 = -3x - 15 \implies 4x = -8 \implies x = -2

.

When the absolute value expressions have opposite signs,

|x - 7| = 3|x + 5|

simplifies to

x - 7 = -3(x + 5)

.

4

Calculate the sum of all possible values of

x

(-11) + (-2) = -13

.

Summing the two solutions gives the final required value.

Key Concept

Distance on a number line and absolute value equations

Estimated Time:1m 30s

Question 18Question →

If

x

is a positive integer such that

\sqrt{4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1}} = 30,000

what is the value of

x

?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Factoring the common exponential term

4^x \cdot 5^{2x}

inside the radical yields

\sqrt{(4^x \cdot 5^{2x})(4 + 5)} = \sqrt{9 \cdot 4^x \cdot (5^2)^x} = \sqrt{9 \cdot (4 \cdot 25)^x} = \sqrt{9 \cdot 100^x} = \sqrt{9 \cdot 10^{2x}} = 3 \cdot 10^x

. Setting

3 \cdot 10^x = 30,000

gives

10^x = 10,000 = 10^4

, which means

x = 4

.

Step-by-Step Solution

1

Separate the addition in exponents using exponent rules.

4^{x+1} \cdot 5^{2x} = 4^x \cdot 4^1 \cdot 5^{2x}

and

4^x \cdot 5^{2x+1} = 4^x \cdot 5^{2x} \cdot 5^1

.

Applying the product rule of exponents

a^{m+n} = a^m \cdot a^n

prepares terms for factoring.

2

Factor out the common expression

4^x \cdot 5^{2x}

from the sum inside the radical.

4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1} = (4^x \cdot 5^{2x})(4 + 5) = 9 \cdot 4^x \cdot 5^{2x}

.

Factoring converts the sum under the square root into a single product.

3

Combine terms with powers into base 10.

4^x \cdot 5^{2x} = 4^x \cdot (5^2)^x = 4^x \cdot 25^x = (4 \cdot 25)^x = 100^x = 10^{2x}

.

Using power of a power

(a^m)^n = a^{mn}

and power of a product

a^n b^n = (ab)^n

simplifies the expression into powers of 10.

4

Take the square root of the simplified product.

\sqrt{9 \cdot 10^{2x}} = \sqrt{9} \cdot \sqrt{10^{2x}} = 3 \cdot 10^x

.

Applying the product rule for radicals

\sqrt{ab} = \sqrt{a}\sqrt{b}

and halving the exponent

(10^{2x})^{1/2} = 10^x

.

5

Equate the simplified expression to 30,000 and solve for

x

.

3 \cdot 10^x = 30,000 \implies 10^x = 10,000 \implies 10^x = 10^4 \implies x = 4

.

Dividing both sides by 3 isolates

10^x

, and matching exponential bases gives

x = 4

.

Key Concept

Exponent Rules and Radical Simplification

Estimated Time:2m 0s

Question 19Question →

If $x$ and $y$ are real numbers such that $|x + 2| \le 3$ and $|y - 5| \le 2$ , which of the following could be the value of $|x - y|$ ? Select all such values.

Select all that apply

1

4

8

12

15

Show answer & explanation

Answer: 4; 8; 12

Answer

The possible values for

|x - y|

are 4, 8, and 12.

Solving the inequality

|x + 2| \le 3

gives

-5 \le x \le 1

, and solving

|y - 5| \le 2

gives

3 \le y \le 7

. The minimum possible value of

x - y

occurs at

-5 - 7 = -12

, and the maximum value occurs at

1 - 3 = -2

. Thus,

x - y

lies entirely in the interval

[-12, -2]

. Taking absolute values shows that

|x - y|

must lie in the interval

[2, 12]

. The numbers 4, 8, and 12 all fall within this interval and are valid solutions.

Step-by-Step Solution

1

Solve the absolute value inequality for

x

.

-3 \le x + 2 \le 3 \implies -5 \le x \le 1

Unpack

|x + 2| \le 3

into a compound inequality and isolate

x

.

2

Solve the absolute value inequality for

y

.

-2 \le y - 5 \le 2 \implies 3 \le y \le 7

Unpack

|y - 5| \le 2

into a compound inequality and isolate

y

.

3

Determine the minimum and maximum possible values of

x - y

.

Minimum

x - y = -5 - 7 = -12

; Maximum

x - y = 1 - 3 = -2

.

To minimize

x - y

, take the smallest

x

and largest

y

. To maximize

x - y

, take the largest

x

and smallest

y

.

4

Find the range of

|x - y|

.

2 \le |x - y| \le 12

Since

-12 \le x - y \le -2

, taking the absolute value yields values in the closed interval

[2, 12]

.

5

Evaluate the choices against the interval

[2, 12]

.

4, 8, and 12 lie inside

[2, 12]

, whereas 1 and 15 do not.

Any real number in

[2, 12]

is a achievable value for

|x - y|

.

Key Concept

Real Numbers, Number Line, and Absolute Value Inequalities

Question 20Question →

Suppose $a$ , $b$ , and $c$ are integers such that $a < 0 < b < c$ . If $a(b - c)$ is an odd integer and $a^2 b + b c$ is an even integer, which of the following expressions MUST be a positive even integer?

b(c - a)

a b c

a(b - c)

(a - c) b

(a + c) b

Show answer & explanation

Answer: $b(c - a)$

Answer

b(c - a)

is guaranteed to be a positive even integer.

The expression

b(c - a)

consists of

b

(which is positive and even) multiplied by

(c - a)

(which is positive and even, as subtracting a negative odd number from a positive odd number yields a positive even number). The product of two positive even integers is always a positive even integer.

Step-by-Step Solution

1

Determine the signs of variables

a

,

b

, and

c

.

a < 0

(negative),

b > 0

(positive), and

c > 0

(positive).

Directly given by the inequality

a < 0 < b < c

.

2

Analyze parity from the given condition that

a(b - c)

is odd.

a

is odd and

(b - c)

is odd.

A product of two integers is odd if and only if both factors are odd.

3

Analyze parity from the second condition

a^2 b + b c = b(a^2 + c)

being even.

b

must be even, and

c

must be odd.

Since

a

is odd,

a^2

is odd. If

b

were odd, then

a^2 + c

would need to be even (making

c

odd), but if both

b

and

c

were odd,

b - c

would be even, contradicting step 2. Therefore,

b

must be even. Since

b

is even and

b - c

is odd,

c

must be odd.

4

Evaluate the sign and parity of

b(c - a)

.

c - a = \text{odd} - (\text{negative odd}) = \text{positive even}

. Since

b

is positive even,

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

.

Subtracting a negative number yields addition (

c - a > 0

), and subtracting an odd integer from an odd integer produces an even integer.

Key Concept

Even-Odd Parity and Integer Sign Properties under Multiplication and Subtraction

Arithmetic

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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