### Instruction: How many positive integers, including $1,$ are divisors of both $40$ and $72?$.### Response: I need to find the common divisors of $40$ and $72,$ which are the factors of their greatest common divisor (GCD).
One way to find the GCD of two numbers is to use prime factorization.
I can write $40$ as $2^3 \times 5$ and $72$ as $2^3 \times 3^2.$
The GCD is the product of the common prime factors with the lowest exponent, which is $2^3 = 8.$
So I need to find the factors of $8,$ which are $1, 2, 4,$ and $8.$
There are four of them, so that's the answer.