zisis wrote:
If 2a – b = 3c, where a, b, and c are non-zero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
A. -2
B. -1
C. 1
D. 10
E. 12
Responding to a pm:
Bunuel has already given the algebraic approach which is quite simple and clear. I am guessing that since you are looking for another approach, you want me to solve it without using algebra.
I instinctively jumped to the options in this question. They are the average of a and b so sum of a and b will be twice of the average so depending on which option we pick, the sum (a + b) will be -4 or -2 or 2 or 20 or 24. Note that a and b will be either both even or both odd since their sum must be even.
The question says, which of the following could be the average i.e. there are probably many numbers that could be the average and one of them in included in this list.
We also know that 2a – b = 3c
Since right hand side has a 3, we know that 2a - b is divisible by 3. The easiest way to make it divisible by 3 is to make both a and b divisible by 3 which makes their sum divisible by 3 as well. Of the given options, only 24 is divisible by 3 hence (E) must be the answer.
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