The sum of two positive integers is 60 and their least common multiple is 273 . What are the two integers?

(The least common multiple of two positive integers is the smallest positive integer which is a multiple of these two integers.)
Suppose that the two integers are $m$ and $n$.

We are told that $m+n=60$ and $\operatorname{lcm}(m, n)=273$.

We note that $273=3 \cdot 91=3 \cdot 7 \cdot 13$.

Since 273 is a multiple of each of $m$ and $n$, this means that $m$ and $n$ are both divisors of 273 .

The divisors of 273 are 1, 3, 7, 13, 21, 39, 91, 273.

Of these, the only pair that has a sum of 60 is 21 and 39 .

Thus, the two integers are 21 and 39.

We note that since $21=3 \cdot 7$ and $39=3 \cdot 13$, then an integer is a common multiple of 21 and 39 exactly when it has prime factors of 3,7 and 13 , so the smallest such integer is $3 \cdot 7 \cdot 13=273$ which makes $\operatorname{lcm}(21,39)=273$.

ANSWER: 21 and 39