Possible HCF?

If the greatest common divisor of two numbers is 37, both numbers are divisible by 37. $\therefore$ Let the numbers be $a = 37x, b = 37y$ . We are given that the sum of a and b = 629. $\therefore$ $37(x+y) = 629 \implies x + y = 17$ . There are total 16 unordered pairs since both $a, b$ are positive. But since $(a,b)$ is the same as $(b,a)$ . There are half as many pairs $= \boxed{8}$

Let the two numbers be $m$ and $n$ . Since their greatest common divisor is $37$ , which is a prime. Then $m=37a$ and $n=37b$ , where $a$ and $b$ are positive integers. Then $m+n = 37(a+b) = 629$ $\implies a+b = 17$ and there are $\boxed 8$ possible pairs $(1,16)$ , $(2,15)$ , $(3,14)$ , $\cdots$ , $(8,9)$ .