If k is a positive integer that is divisible by both 6 and 10, which of the following expr… — GMAT

If $k$ is a positive integer that is divisible by both $6$ and $10$ , which of the following expressions MUST yield an integer?

The expression

\dfrac{k}{15}

MUST yield an integer.

Since

k

is divisible by both

6

and

10

k

must be a multiple of

\text{LCM}(6, 10) = 30

. Therefore,

k = 30m

for some positive integer

m

. Dividing

k

15

gives

\dfrac{30m}{15} = 2m

, which is guaranteed to be an integer for any positive integer

m

Determine the least common multiple (LCM) of

6

and

10

\text{LCM}(6, 10) = 30

Since

k

is divisible by both

6

and

10

k

must be a multiple of their LCM.

Express

k

in terms of its smallest possible base multiple.

k = 30m

for some positive integer

m

This represents all possible integer values for

k

Determine which choice has a denominator that divides

30

Since

15

divides

30

\dfrac{k}{15} = \dfrac{30m}{15} = 2m

, which is always an integer.

Any divisor of

30

will divide

k

for all valid values of

k

Least Common Multiple (LCM) and Divisibility Properties

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