Can You Find the GCD?
As ranged over all positive integers, what is/are the value(s) of
Notation : denotes the greatest common divisor function.
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1 solution
Relevant wiki: Greatest Common Divisor - Problem Solving
Its not possible for any a to make the denominator and numerator divisible by 2,3 or 5. Lets assume it is. So we define P = { 2 , 3 , 5 } , p ∈ P and assume that ∃ p ∈ P : p ∣ a 2 + a + 1 ∧ p ∣ a 2 − a − 1 . If a number divides two others, it also divides their sum and difference. So we conclude ∃ p ∈ P : p ∣ 2 a 2 ∧ p ∣ 2 a + 2 . So we assume p divides a product, and because p is prime, it has to occur in the prime factorization of the Product. Therefor it has to divide at least one of the factors. Thus p = 2 divides a and a + 1 at the same time, what obviously isnt possible. And as the numerator and denominator are always odd, they cant get divided by p = 2 either.So our assumption becomes wrong which led us to the solution, that the fraction cant be reduced with 2,3 or 5.