Question

please help me prove this:For any integers a, 3 divides a(a+1)(a+2).

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Answer 1

Dividend= divisor x quotient + remainder
a=3q + r
here 0<r<3 and r belongs to integer. Hence r=1 or r=2
for r=1
a=3q+1
so a+2=3q+3
since 3q+3 is multiple of 3, a+2 should also be a multiple of 3
Now consider r=2
so a=3q+2
a+1=3q+3
which is again divisible by 3
so for any integer a,
either a is divisible by 3 or a+1 is divisible by 3 or a+2 is divisible by 3
hence a(a+1)(a+2) is always divisible by 3.

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