Quandaries
and Queries
What is the probability that a randomly chosen 3 digit number has exactly 3 factors
Hi,
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
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.
Penny