The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

How many triples of integers a, b, c satisfy:

There are three unknowns but they are all integer. The equation is Diophantine.

Use the fundamental theorem of arithmetic (any integer has a unique prime factorization). This means that, if we factor the right hand side into primes, then the left hand side must have the same prime factorization:

Since 2 and 3 are not perfect squares, the only values possible for c+4 are:

Case 1 (c+4=1): Since 2b+1 is necessarily an odd number, then it can only have the values:

and the corresponding values of a are:

Case 2 (c+4=11): Since 2b+1 is necessarily an odd number, then it can only have the values:

There are, therefore, 4 possible triples: