Someone said in the summer: "My birth year is divisible by my current age." A friend replied: "Although I am older than you, I can still say the same about myself." - How old are the speakers?
One of the conversationalists' age is denoted by $x$, and their birth year is $1961-x$. This is divisible by $x$ if and only if 1961 is divisible by it. Therefore, 1961 is divisible by both conversationalists' ages.

The factorization of 1961 is $1961=37 \cdot 53$, and these factors cannot be broken down further. Thus, the divisors of 1961 are: 1, 37, 53, 1961. The non-trivial divisors in our case are unacceptable, so the conversationalists are 37 and 53 years old.