SOLUTION: Prove or disprove algebraically: The sum of an even number and an odd number is odd.

Question 1096698: Prove or disprove algebraically: The sum of an even number and an odd number is odd.
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Prove or disprove algebraically: The sum of an even number and an odd number is odd.
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For any integer n:
2n is even
2n+1 is odd
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2n + 2n+1 = 4n+1 --> odd

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

k = any integer
2*k = any even integer
2*k+1 = any odd integer

Add up the even number (2*k) and the odd number (2*k+1) to get the following:
( 2*k ) + ( 2*k+1 )
2*k + 2*k+1
(2*k + 2*k)+1
(2+2)*k+1
4*k+1

Then we can rewrite that last expression like so:
4*k+1
(2*2)*k+1
2*(2*k)+1

If we let m = 2*k, then we have the expression 2*m+1 which is an odd number (since m is also an integer)

Therefore the initial claim The sum of an even number and an odd number is odd is always true. This concludes the proof.

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