SOLUTION: Suppose that you square two consecutive whole numbers and subtract the smaller square from the larger. Is it possible that the difference is an even number? Explain your answer wi

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Question 184232: Suppose that you square two consecutive whole numbers and subtract the smaller square from the larger. Is it possible that the difference is an even number? Explain your answer with appropriate examples.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let k=first whole number, (ie k is NOT a fraction or decimal number)

So k+1 would be the next consecutive whole number


Now square k%2B1 to get %28k%2B1%29%5E2=k%5E2%2B2k%2B1

Now subtract k%5E2 (the square of the first number) from k%5E2%2B2k%2B1 (the square of the second) to get

%28k%5E2%2B2k%2B1%29-k%5E2=%28k%5E2-k%5E2%29%2B2k%2B1=2k%2B1


Now remember, we said at the top that "k" is a whole number. So this means that 2k%2B1 is guaranteed to be an odd number (since 2k is an even integer)


So it is NOT possible to get an even difference between the squares of two consecutive whole numbers.


Examples (these don't prove the statement above, but help show it)

ex 1: Pick a number 6 and the next number 7. Square 6 to get 36. Square 7 to get 49

Subtract: 49-36=13

The difference is odd


ex 1: Select the number 12 and the next number 13. Square 12 to get 144. Square 13 to get 169

Subtract: 169-144=25

The difference is odd

You can try any two values and you'll find that the difference is odd.