21k + 75i = 3(7k+25i) with k,i in Z. Therefore 7k+25i is in Z and thus each element of the form 21k + 75i can be written as an element of the form 3p where p is an integer. Hence [21,75] is a subset of [3].

By Bezout's Identity and gcd(21,75) = 3, it follows that there exists an n,m in Z such that 21n + 75m = 3.
So 21kn + 75km = 3k with kn and km in Z. Therefore every element of the form 3k can be written as an element of the form 21(kn) + 75(km) where kn,km are integers. Hence [3] is a subset of [21,75]

Since they are both subsets of each other, it follows that [21,75] = [3]