number properties' puzzle (not proved?)

Question

If m and r are integers divisible by 5 and m>n, which of the following is not necessarily true?
a) (m-n) is divisible by 5
b) (m^2-n^2) is divisible by 25
c) (m+n) is divisible by 10
d) (m+n) is divisible by 12
e) all of these

OA is (e)
The explanation of the solution method if the following: 
the question can be solved for any integers divisible by 5. Let m=8 and n=7, then
(m-n)=(8-7)=1, which is not divisible by 5;
(m^2-n^2)=(64-49)=15, which is not divisible by 25;
(m+n)=(8+7)=15, which is not divisible by 10 or 12

BUT I cannot understand this explanation because the question states that “m and r are integers divisible by 5” and to solve the question it is assumed that m=8 and n=7. If we take it for granted then we may also suppose the correctness of (m+n)=(8+7)=15. 
In other words, if 8 and 7 are numbers divisible by 5 as it is given in the explanation (the solution will not be an integer for both numbers, i.e. 8/5=1.6 and 7/5=1.4 ) then (m+n)=(8+7)=15 must be allowed to be divisible by 10 or 12. Hence: (c) and (d) must be necessarily true.

Pls., give me alternative explanations to disprove my remark?

shrouded1 · Answer

something looks very wrong about this question. what is the source ?

gurpreetsingh · Answer

I think A and B both satisfies.

very bad question.

feruz77 · Answer

The source is The Math Question Bank of the WinningPrep.com

IanStewart · Answer

Yes, this question and the explanation you quoted are nonsensical. m cannot be 8, because 8 is not divisible by 5.

The question is strange to begin with, because it tells us m and  r  are divisible by 5, but r is not mentioned anywhere else in the question. I assume they mean to tell us that m and  n  are divisible by 5. In that case, m = 5q and n = 5r, for some integers q and r, where q > r.

So:
I) m-n = 5q - 5r = 5(q-r), and m-n certainly is divisible by 5.
ii) m^2 - n^2 = (m+n)(m-n) = (5)(q+r)(5)(q-r) = 25*(q^2 - r^2), and m^2 - n^2 certainly is divisible by 25.
iii) m+n could be equal to 10 + 5 = 15, so m+n is not necessarily divisible by 10.
iv) again, m+n could be 15, so is not necessarily divisible by 12.

So both C and D are not necessarily true, and there's no proper way to answer the question; it has two correct answers.

number properties' puzzle (not proved?)

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