document.write( "Question 1182800: How many pairs (a,b) of positive integers are there such that a and b are factors of 6^6 and a is a factor of b? \n" ); document.write( "

Algebra.Com's Answer #812951 by robertb(5830) $\"\"$ $\"About$

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I will have to go with ikleyn's solution here.\r
\n" ); document.write( "\n" ); document.write( "The problem with greenestamps' solution is that he essentially assumes that a*b = 6^6, which is an incorrect assumption.\r
\n" ); document.write( "\n" ); document.write( "ikleyn's solution is correct, but the solution requires a bit of motivation combinatorially, especially when explaining
\n" ); document.write( "(1+2+3+4+5+6+7)*(1+2+3+4+5+6+7) = 28^2 = 784.\r
\n" ); document.write( "\n" ); document.write( "In fact the problem can be generalized further to the number A^n*B^m and not only to 2^6*3^6.
\n" ); document.write( "(So the bases 2 and 3 are irrelevant, and A and B can be any distinct positive integers both greater than 1.)\r
\n" ); document.write( "\n" ); document.write( "In this case the number of ordered pairs will be ((n+1)(n+2))/2*((m+1)(m+2))/2 = (n+1)(n+2)(m+1)(m+2)/4. \r
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