document.write( "Question 845434: prove sum of the cubes of two numbers is odd, iff one number is even and one is idd \n" ); document.write( "

Algebra.Com's Answer #509285 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2N\"$ is an even number for any non-zero $\"N\"$ .
\n" ); document.write( "Look at its cube.
\n" ); document.write( " $\"%282N%29%5E3=8N%5E3=2%284N%5E3%29\"$ . So the cube of an even number is also an even number.
\n" ); document.write( " $\"2N%2B1\"$ is an odd number.
\n" ); document.write( "Look at its cube.
\n" ); document.write( " $\"%282N%2B1%29%5E3=8N%5E3%2B12N%5E2%2B16N%2B1=2%284N%5E3%29%2B2%286N%5E2%29%2B2%288N%29%2B1\"$
\n" ); document.write( "So the cube of an odd number is the sum of 3 even terms and 1 so that makes it odd.
\n" ); document.write( "The sum of an even number and an odd number is odd so therefore the sum of the cubes of an even number and an odd number is also odd.
\n" ); document.write( "If both numbers were even, the sum of the cubes would be even.
\n" ); document.write( "If both numbers were odd, the sum of the cubes would be even because the 1+1 terms would make it even.
\n" ); document.write( "So the sum of cubes is odd iff one number is even and the other is odd. \n" ); document.write( "

\n" );