Question 1129675: Prove that the value of the expression is not divisible by 6 for any whole n:
(2n+1)(n+5)–2(n+3)–(5n+13). Found 2 solutions by MathLover1, greenestamps:Answer by MathLover1(20849) (Show Source):
You can put this solution on YOUR website!
Hint: an integer is divisible by if and only if it is divisible by both and .
Can you show that your expression is even,divisible by both and ?
first simplify it:
=> GFC is ;
so,it is divisible by ,
but it is divisible by
=>consequently, it is divisible by either
The proof by the other tutor is not complete. She shows that the given expression in simplified form is equivalent to
and then states without proof that the expression is divisible by 2 but not by 3.
But to complete the proof that the expression is not divisible by 6 for ANY whole number, we have to prove that the factor is NEVER divisible by 3.
We do that with modular arithmetic.
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 is not divisible by 3 -- that is, that it is not equal to 0 mod 3.
(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)
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.
