1954 AHSME Problems/Problem 49
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The difference of the squares of two odd numbers is always divisible by 
. If 
, and 
and 
are the odd numbers, to prove the given statement we put the difference of the squares in the form:
![$\textbf{(A)}\ (2a+1)^2-(2b+1)^2\\ \textbf{(B)}\ 4a^2-4b^2+4a-4b\\ \textbf{(C)}\ 4[a(a+1)-b(b+1)]\\ \textbf{(D)}\ 4(a-b)(a+b+1)\\ \textbf{(E)}\ 4(a^2+a-b^2-b)$](http://latex.artofproblemsolving.com/f/a/7/fa77929520ec7fa33eff255dfa3b46759f3d0c8a.png)
Solution
Although all of the forms listed can be used to show that the difference of 
and 
is necessarily divisible by , we should use 
because then the second and third factors are necessarily of different parities, so that their product is necessarily even and we are done.
See Also
| 1954 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 48 |
Followed by Problem 50 | |
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