Let n > 1 be an integer. A relation R on the set of natural numbers N is defined as R = (x, y) : 37x + 83y is a multiple of n . The number of possible values of n for which R is an equivalence relation is

Step-by-step solution

For the relation R to be an equivalence relation, it must be reflexive. For R to be reflexive, xRx must hold for all x N . This implies that 37x + 83x must be a multiple of n for all x N . 120x is a multiple of n for all x N . For this to be true for x = 1 , n must be a divisor of 120 . If n divides 120 , then 83 -37 n . The condition 37x + 83y 0 n becomes 37x - 37y 0 n , or 37(x - y) 0 n . This relation is clearly symmetric and transitive for any n dividing 120 . Thus, n can be any divisor of 120 strictly greater