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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:

equation

which of the following statements can be true?

It's always a good idea to factor integers when they come up in problems:

equation

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.

Therefore, since:

equation

m is either 2 or -2 since its exponent is even and makes the result positive.

Also:

equation

is another Diophantine equation.

Factor the right hand side into primes:

equation

The only possible choices that satisfy the requirements are:

equation

where pairs are matched accordingly.

Regardless how we choose the pairs, we have:

equation

Therefore, I, III and IV can be true.