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.
If m is integer and a,b,c,d are distinct positive integers greater than 1 and:
which of the following statements can be true?
It's always a good idea to factor integers when they come up in problems:
This is a Diophantine equation in factored form. According to the fundamental theorem of arithmetic, the prime factorization has to be the same on both sides of the equation.
m is either 2 or -2 since its exponent is even and makes the result positive.
is another Diophantine equation.
Factor the right hand side into primes:
The only possible choices that satisfy the requirements are:
where pairs are matched accordingly.
Regardless how we choose the pairs, we have:
Therefore, I, III and IV can be true.