document.write( "Question 1192007: Prove:
\n" ); document.write( "if 3 is not divisible by m * n, then 3 is not divisible by m and 3 is not divisible by n \n" ); document.write( "

Algebra.Com's Answer #823900 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Proof by contradiction:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Assume that 3 is a factor of m. We'll show a contradiction arises from this assumption.
\n" ); document.write( "This means m = 3k for some integer k
\n" ); document.write( "This further leads to mn = 3k*n = 3(kn)
\n" ); document.write( "Showing that mn is a multiple of 3.
\n" ); document.write( "But this contradicts the fact that mn is not a multiple of 3.
\n" ); document.write( "Therefore, we must have m be a non-multiple of 3 as well.\r
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\n" ); document.write( "\n" ); document.write( "Similar steps would apply to show that n must be a non-multiple of 3. This is one application of \"without loss of generality\" (WLOG) you can do.\r
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\n" ); document.write( "\n" ); document.write( "Ultimately you should find that we get a contradiction if either m = 3k or n = 3p for integers k and p. Therefore, m and n cannot be multiples of 3.
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