document.write( "Question 1129675: Prove that the value of the expression is not divisible by 6 for any whole n:
\n" ); document.write( "(2n+1)(n+5)–2(n+3)–(5n+13). \n" ); document.write( "

Algebra.Com's Answer #746282 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "The proof by the other tutor is not complete. She shows that the given expression in simplified form is equivalent to

\n" ); document.write( " $\"2%28n%5E2%2B2n-7%29\"$

\n" ); document.write( "and then states without proof that the expression is divisible by 2 but not by 3.

\n" ); document.write( "But to complete the proof that the expression is not divisible by 6 for ANY whole number, we have to prove that the factor $\"n%5E2%2B2n-7\"$ is NEVER divisible by 3.

\n" ); document.write( "We do that with modular arithmetic.

\n" ); document.write( "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 $\"n%5E2%2B2n-7\"$ is not divisible by 3 -- that is, that it is not equal to 0 mod 3.

\n" ); document.write( "(1) n = 0 mod 3: n^2+2n-7 mod 3 = 0+0-7 = -7 mod 3 (not = 0 mod 3)
\n" ); document.write( "(2) n = 1 mod 3: n^2+2n-7 mod 3 = 1+2-7 = -4 mod 3 (not = 0 mod 3)
\n" ); document.write( "(3) n = 2 mod 3: n^2+2n-7 mod 3 = 1+1-7 = -5 mod 3 (not = 0 mod 3)

\n" ); document.write( "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. \n" ); document.write( "

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