Let a and b be positive integers such that 15a = 28b. Which of the following statements MU… — GRE General Test

Let $a$ and $b$ be positive integers such that $15a = 28b$ . Which of the following statements MUST be true? Select all that apply.

$a$ is divisible by $7$ Answer
$b$ is divisible by $15$ Answer
C
$a$ must be an odd integer
D
$b$ must be a prime number
E
$a + b$ must be divisible by $30$

Answer

The statements asserting that

a

is divisible by

7

and that

b

is divisible by

15

must be true.

Equating prime factorizations gives

(3 \cdot 5) \cdot a = (2^2 \cdot 7) \cdot b

. Because

15

and

28

share no common prime factors,

a

must contain

2^2 \cdot 7 = 28

as a factor, and

b

must contain

3 \cdot 5 = 15

as a factor. Therefore,

a

is divisible by

7

(since

7

divides

28

) and

b

is divisible by

15

Step-by-Step Solution

Analyze the prime factorization of both sides of the equation.

The given equation

15a = 28b

can be rewritten as

(3 \cdot 5) \cdot a = (2^2 \cdot 7) \cdot b

Prime factorizations allow us to identify fundamental divisibility relationships between variables.

Determine the prime factors required for

a

and

b

a

must be divisible by

28

(i.e.,

a = 28k

for some integer

k \ge 1

), and

b

must be divisible by

15

(i.e.,

b = 15k

for the same integer

k

Since

\gcd(15, 28) = 1

, all prime factors of

28

(

2^2

and

7

) must come from

a

, and all prime factors of

15

(

3

and

5

) must come from

b

Evaluate each choice against these general forms.

Since

a = 28k

a

is a multiple of

7

, making the first statement true. Since

b = 15k

b

is a multiple of

15

, making the second statement true.

Any factor of

28

is automatically a factor of

a

, and any factor of

15

is automatically a factor of

b

Key Concept

Fundamental Theorem of Arithmetic and Coprime Divisibility Rules

Question

Answer

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Key Concept

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