If a and b are positive integers such that 15a is divisible by 24 and 10b is divisible by … — GRE General Test

If $a$ and $b$ are positive integers such that $15a$ is divisible by $24$ and $10b$ is divisible by $14$ , which of the following statements MUST be true? Select all such statements.

$a$ is divisible by $8$ Answer
B
$b$ is divisible by $14$
The product $ab$ is divisible by $56$ Answer
D
The sum $a + b$ must be an even integer
E
$a$ is divisible by $24$

Answer

The statements asserting that

a

is divisible by

8

and that the product

ab

is divisible by

56

must be true.

Because

24 = 2^3 \times 3

and

15 = 3 \times 5

a

must provide three factors of

2

, meaning

a

is divisible by

8

. Similarly,

14 = 2 \times 7

and

10 = 2 \times 5

, so

b

must provide a factor of

7

, meaning

b

is divisible by

7

. Multiplying the minimum properties (

a

is a multiple of

8

and

b

is a multiple of

7

) guarantees that

ab

is a multiple of

56

Step-by-Step Solution

Analyze the condition

15a

is divisible by

24

Prime factorizations:

15 = 3 \times 5

and

24 = 2^3 \times 3

. For

24

to divide

15a

a

must supply

2^3 = 8

, so

a

is a multiple of

8

Determines the prime factors required for

a

Analyze the condition

10b

is divisible by

14

Prime factorizations:

10 = 2 \times 5

and

14 = 2 \times 7

. For

14

to divide

10b

b

must supply

7

, so

b

is a multiple of

7

Determines the prime factors required for

b

Evaluate each statement against the deduced minimum properties of

a

and

b

Since

a

is a multiple of

8

, the statement regarding

a

being divisible by

8

is always true. Since

a

is a multiple of

8

and

b

is a multiple of

7

ab

is a multiple of

8 \times 7 = 56

. Counterexamples disprove the remaining choices (

b=7

disproves divisibility of

b

14

and evenness of

a+b

;

a=8

disproves divisibility of

a

24

Identifies which statements MUST be true for all valid values of

a

and

b

Key Concept

Divisibility and Prime Factorization Constraints

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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