Bunuel 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.
Kudos for a correct solution.
800score Official Solution:(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.
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