document.write( "Question 9579: If x is an integer, x is even if x=2p for some integer p, and x is odd if for no integer p such that x=2p. If ab is an odd number, show that a and b are both odd. \n" ); document.write( "

Algebra.Com's Answer #5212 by prince_abubu(198) $\"\"$ $\"About$

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If a is an odd number, it has to be expressed as a = 2m + 1 for ANY integer m. The 2m forces it to be even, the +1 forces the expression to be odd. Similarly, b = 2n + 1 for some integer n. Now, we're going to prove that a*b is odd:\r
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\n" ); document.write( "\n" ); document.write( " $\"+ab+=+%282m+%2B+1%29%282n+%2B+1%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"+ab+=+4mn+%2B2m+%2B+2n+%2B+1+\"$ <---------- The product 4mn is even no matter what. So are 2m + 2n. The +1 throws off the \"evenhood\", so a*b is odd. \n" ); document.write( "

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