Let S be the set of all integers n in the interval [100, 500] such that the greatest common divisor of n and 21 is 1 . The number of elements in S is

Step-by-step solution

The condition (n, 21) = 1 means that n is coprime to 21 . Since 21 = 3 7 , n must be neither a multiple of 3 nor a multiple of 7 . First, find the total number of integers in the interval [100, 500] : Total = 500 - 100 + 1 = 401 . Let A be the set of multiples of 3 in [100, 500] and B be the set of multiples of 7 in [100, 500] . For set A (multiples of 3 ): The first multiple is 102 and the last is 498 . Number of terms n(A) = 498 - 102 3 + 1 = 132 + 1 = 133 . For set B (multiples of 7 ): The first multiple is 105