### Instruction: How many positive integers, including $1,$ are divisors of both $40$ and $72?$.### Response: To find the divisors of both $40$ and $72,$ I need to find their common factors.
A common factor of two numbers is a factor of their greatest common divisor (GCD).
So I need to find the GCD of $40$ and $72.$
One way to do that is to use the prime factorization of each number and look for the common prime factors with the lowest powers.
The prime factorization of $40$ is $2^3 \times 5.$
The prime factorization of $72$ is $2^3 \times 3^2.$
The common prime factors are $2^3.$
So the GCD of $40$ and $72$ is $2^3 = 8.$
Now, to find the divisors of $8,$ I need to consider all the possible combinations of its prime factors.
The prime factorization of $8$ is $2^3.$
So the possible combinations are $1, 2, 2^2, 2^3.$
These are the only positive integers that are divisors of both $40$ and $72.$
There are four of them.