bettatantalo wrote:

If \(a\) is a positive integer, then \((a+2)(a+3)(a+4)(a+5)\) is

A) even only when \(a\) is odd

B) odd whenever \(a\) is odd

C) divisible by 3 only when \(a\) is odd

D) divisible by 4 whenever \(a\) is even

There is no E) choice in this question I got given by my tutors.

Language in the correct answer has a trap.

Test numbers. What happens to

(a+2)(a+3)(a+4)(a+5)

-- When

a is odd?

-- When

a is even?

ODD: When

a=1 then the first factor is

(a + 2) = 3, so

when

a is odd, (3 * 4 * 5 * 6) = 360

EVEN: When

a=2, the first factor is 4.

When

a is even, (4 * 5 * 6 * 7) = 840

Check options against test numbers

(a+2)(a+3)(a+4)(a+5) is

A) even

only when

a is odd:

FALSE. From test cases:

The product (360, 840) is even when

a is odd AND when

a is even. REJECT

B) odd whenever

a is odd:

FALSE.

When

a is odd, the product (360) is even, not odd. REJECT

C) divisible by 3

only when

a is odd:

FALSEThe product (360, 840) is divisible by 3 when

a is odd AND when

a is even. REJECT

D) divisible by 4 whenever

a is even:

(1) TRUEWhen

a = 2 (even), the product of 840 is divisible by 4. KEEP

If concerned, check

a = 2. First factor (4 + 2) = 6

(6 * 7 * 8 * 9) = 3,024. Divisible by 4. KEEP

(2) "Divisible by 4" is ALSO TRUE when

a is odd

(When

a = 1, product is 360, which is divisible by 4)

Possible trap, but:What happens when a is odd does not matter. #2 does not make #1 false. Option D is not asking about #2.*

Option D does not state that the product is divisible by 4

only when a is even.

D asks about the result

when a is even.

Well, when

a is even, the product of the factors is indeed divisible by 4.

Question asked, question answered.

Answer D

*In option D, we don't care about what happens in cases other than a = even.

Stay with what is asked. If or when a is even, is the product divisible by 4? Yes?

Then forget about what happens when a is odd. We are not being asked about that situation.

If D had stated "only when a is even," we would have a different scenario. We would have to account for what happens when ais odd.

-- ONLY WHEN implies: true for this situation AND NOT for other situations

Those "other situations" = a is odd

-- WHENEVER implies: true for this situation

Other situations do not matter.
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