For any integer k, the expression k(k + 1)(k + 5) must be an even integer. — GMAT

For any integer $k$ , the expression $k(k + 1)(k + 5)$ must be an even integer.

Answer: Answer

True. The expression

k(k + 1)(k + 5)

is guaranteed to be an even integer for all integer values of

k

Because

k

and

k + 1

are consecutive integers, one of them must be even. The product of an even integer and any other integer is always even, making

k(k + 1)(k + 5)

even for every integer

k

Analyze the parity of the consecutive terms

k

and

k + 1

One of the two integers

k

k + 1

is always even regardless of whether

k

is even or odd.

Consecutive integers always alternate between even and odd.

Determine the parity of the product of an even integer and any other integer.

The product

k(k + 1)

is always an even integer.

The product of an even integer and any integer is always even.

Evaluate the full expression

k(k + 1)(k + 5)

Since

k(k + 1)

is even, multiplying by

(k + 5)

produces an even integer.

An even integer multiplied by any integer results in an even integer.

Parity of products of consecutive integers

Estimated Time:1m 0s

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