Burt Kaliski Jr. (@modulomathy)
3/20/12 11:06 AM
#math trivia for #March20: #80 has the form x^4-1 for x > 2 so must have at least three nontrivial divisors. What are they, and why three?
The nontrivial divisors of 80 (those other than 80 or 1) are 2, 4, 5. 8, 10, 16, 20, and 40.
The reason there are at least three is that x^4-1 can be expressed as the product
x^4-1 = (x^2+1)(x+1)(x-1) .
When x > 2, all three factors have distinct values greater than 1, giving the product at
least three nontrivial divisors. With x = 3 in the present example, the expression becomes
80 = 10*4*2 .
These are not the only divisors, and in general there must be at least six, considering also the products of pairs of factors. When x is odd, there will be more than six because of the various multiples of 2 involved. When x is even, the minimum of six can be achieved exactly, for instance with x = 4.