Which of the following MUST yield an integer when . . . .

Question

Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

The OA is the option A.

How can I show that the correct answer is option A? Experts, could you give me some help, please?

EconomistGMATTutor · Answer

Hello Vincen.

Let's take a look at your question.

The sum of five consecutive positive integers is:

n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10 = 5(n+2).

So, it is multiple of 5. Therefore, it MUST yield an integer when divided by 5.

Hence, there is no need to try the rest of the options.

The correct answer is A.

I hope this explanation may help you.

Regards.

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If you want we can prove that the rest of the options are incorrect.

Option 2: The square of a prime number is not always divisible by 5. We can choose 3, then 3^2=9.

Option 3: The sum of two odd integers is not always divisible by 5. We can choose 3 and 5, then 3+5=8.

Option 4: The product of three consecutive odd numbers is not always divisible by 5. We can choose 17, 19 and 21.

Option 5: The difference between a multiple of 8 and a multiple of 3 is not always divisible by 5. We can choose 24-3=21.

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Scott@TargetTestPrep · Answer

Vincen wrote:Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

The OA is the option A.

How can I show that the correct answer is option A? Experts, could you give me some help, please?

The sum of 5 consecutive integers will always be divisible by 5. If we denote the integers by n, n + 1, … , n + 4; then the sum is 5n + 10, which is divisible by 5 regardless of the number we choose for n. (Note: In fact, the sum of k consecutive positive integers is always divisible by k. So the sum of 5 consecutive positive integers is divisible by 5.)

Alternate Solution:

We can rule out every answer choice except A:

- The square of a prime number is not divisible by 5 unless the prime we choose is 5.
- The sum of two odd integers need not be divisible by 5 (for instance, 1 + 3 = 4 is not divisible by 5).
- The product of three odd integers need not be divisible by 5 (for instance, 9 x 11 x 13 is not divisible by 5).
- The difference between a multiple of 8 and a multiple of 3 need not be divisible by 5 (for instance, 16 - 3 = 13 is not divisible by 5).

Answer: A

Scott Woodbury-Stewart
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