How many positive integers, including $1,$ are divisors of both $40$ and $72?$
The positive integers that divide exactly into $40$ are $1,$ $2,$ $4,$ $5,$ $8,$ $10,$ $20,$ $40.$

The positive integers that divide exactly into $72$ are $1,$ $2,$ $3,$ $4,$ $6,$ $8,$ $9,$ $12,$ $18,$ $24,$ $36,$ $72.$

The numbers that occur in both lists are $1,$ $2,$ $4,$ $8,$ or $\boxed{\mbox{four}}$ numbers in total.