Quandaries and Queries


What is the probability that a randomly chosen 3 digit number has exactly 3 factors



This is a nice problem. The solution depends on determining which positive integers have exactly three factors. Every integer larger that 1 has at least two factors, itself and 1. Suppose that n is an integer, larger than 1 and a is a factor of n which is not n or 1. Then

n = a b

for some integer b which is also not n or 1. Thus 1, n, a and b are factors of n. If n has only three factors then it must be that a = b. Hence

n = a a = a2

and n is a square.

Furthermore, if c is a factor of a then c is a factor of n and hence c must be 1, n or a. Thus a is a prime.



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