Question 1129675
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The proof by the other tutor is not complete.  She shows that the given expression in simplified form is equivalent to<br>
{{{2(n^2+2n-7)}}}<br>
and then states without proof that the expression is divisible by 2 but not by 3.<br>
But to complete the proof that the expression is not divisible by 6 for ANY whole number, we have to prove that the factor {{{n^2+2n-7}}} is NEVER divisible by 3.<br>
We do that with modular arithmetic.<br>
For any whole number n, there are 3 possible values for n, mod 3.  We need to show that in all three cases the factor {{{n^2+2n-7}}} is not divisible by 3 -- that is, that it is not equal to 0 mod 3.<br>
(1) n = 0 mod 3: n^2+2n-7 mod 3 = 0+0-7 = -7 mod 3 (not = 0 mod 3)
(2) n = 1 mod 3: n^2+2n-7 mod 3 = 1+2-7 = -4 mod 3 (not = 0 mod 3)
(3) n = 2 mod 3: n^2+2n-7 mod 3 = 1+1-7 = -5 mod 3 (not = 0 mod 3)<br>
The proof is now complete, because we have shown that, although the expression is always divisible by 2, it is never divisible by 3; and therefore it is never divisible by 6.