Perfect dozens

Number Theory Level 3

If $a, b, c,d$ are integers, what is the remainder when $(a-b) \times (a-c) \times (a-d) \times (b-c) \times(b-d) \times (c-d)$ is divided by 12?

-1 1 -10 -2 11 9 0

1 solution

Kushal Bose
Dec 28, 2016

The four integers $a,b,c,d$ can have different parity of odd and even.There always exists two same parity (odd and even) so, there difference will create two factors of $2$ .So, the above product will always divisible by 4.

Any integer can always be $3k$ or $3k+1$ or $3k+2$ .Here are four integers so at least two integers will be same form whose difference will be divisible by $3$ .

So the above product will be divisible by $3 \times 4=12$

Perfect dozens

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