Total number of 3-digit numbers, whose g.c.d with 36 is 2 , is

Step-by-step solution

Given that (N, 36) = 2 , and 36 = 2^2 3^2 , we require N to be even but not divisible by 4 or 3 . Since 100 N 999 , write N = 2k with k an integer satisfying 50 k 499 . Further, k must be odd to ensure N is not divisible by 4 . The count of odd integers from 50 to 499 is 499 - 51 2 + 1 = 225 . To ensure (N, 36) = 2 , k must not be a multiple of 3 . The number of odd multiples of 3 in this range is 165 - 17 2 + 1 = 75 . Hence, the number of valid k is 225 - 75 = 150 , each giving a unique N = 2k . Final answer: 150