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.

(A) Another way to phrase the question is: Which of the following is divisible by 5?

Choice (A) might take some time to figure out. Others are easier.

(B) is wrong — many squares of primes, such as 2², are not divisible by 5.

(C) is also wrong: consider 3 + 9 = 12.

(D) is incorrect; one example is (7)(9)(11), which is not divisible by 5.

(E) can be shown to be wrong as well: For instance, the difference between 16 and 9 is 7.

Choice (A), then, must be correct. If the first of the five consecutive positive integers is x, the full list is: x + (x +1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10

Since x is an integer, 5x is divisible by 5. 10 is also divisible by 5. Then, 5x + 10 is divisible by 5. -----------

This question could be easier explained, because the sum any five consecutive positive integers could be divided by 5, right?

The given explanation is comprehensive. So the next time you see a similar question you will know many approaches to solve it and can chose the one that will work in that particular case.

The approaches are: 1. To plug in any numbers in the answer choices, until all the choices except one are eliminated. 2. To prove that one of the answer choices satisfies the terms.

NOTE, that if we plug in some specific value into one of the answer choices and the value we get satisfies the terms, it DOES NOT mean that that answer choice is correct. E.g. if we take the answer choice B: The square of a prime number. And plug in x² = 10² = 100. 100 is divisible by 5, but the answer choice is NOT correct for all possible values of x.

So, if to answer your question:

Quote:

This question could be easier explained, because the sum any five consecutive positive integers could be divided by 5, right?

The shortest explanations for each approach could be: 1. (B) is wrong — many squares of primes, such as 2², are not divisible by 5. (C) is also wrong: consider 3 + 9 = 12. (D) is incorrect; one example is (7)(9)(11), which is not divisible by 5. (E) can be shown to be wrong as well: For instance, the difference between 16 and 9 is 7. Therefore the remaining answer choice A must be correct.

2. Let's consider the answer choice A. If the first of the five consecutive positive integers is x, the full list is: x + (x +1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10 Since x is an integer, 5x is divisible by 5. 10 is also divisible by 5. Then, 5x + 10 is divisible by 5. The correct answer is choice A.

Each of these two explanations is sufficient on its own.

Why not option D? If we take 3 consecutive odd number like 3, 5, 7 and product of 3 × 5 × 7 = 105 which is divisible by 5. Why not E? if we take 64 which is 8 multiple and 9 which is 3 multiple, and difference between these two (64 – 9 = 55) which is 55 is divisible by 5.

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

It means that the proper expression MUST be divisible by 5 for ANY plugged in values. You show that they are divisible for some. But are they divisible for all the values? No, they are not, while the answer choice A is.

Hello! Just to clarify, after reading the feedback on the 8th question I still think that there are 2 correct answers to the question, A and E. The reasoning for E is wrong, it has nothing to do with 16 and 9, it should rather be 16 and 6 or 24 and 9, we should use the same multiples, should not we?

E. The difference between a multiple of 8 and a multiple of 3.

This statement does NOT imply that coefficients (or multipliers) in each case must be the same. So a multiple of 8 can be any integer from the set {8, 16, 24, 32, ...} and a multiple of 3 can be any integer from the set {3, 6, 9, 12, ... }. Any pair is possible: (8, 3); (16, 3); (8; 12); ... etc.

If the statement stated that coefficients were the same, then such number would be divisible by 5: n × 8 – n × 3 = n × (8 – 3) = 5n, where n is the coefficient (multiplier).

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