We will find the positive divisors of 14 by finding pairs that multiply to 14.   We begin our list as follows, $$1 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 14.$$  Checking $2$, we find that $2\cdot 7=14$, so our list becomes $$1 \quad 2 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 7 \quad 14.$$  Checking $3$, $4$, $5$, and $6$, we find that none of these are divisors of $14$, so our final list is $$1 \quad 2 \quad 7 \quad 14.$$  Next, we use the buddy method to determine the factors of $42$.  We begin our list as follows, $$1\quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 42.$$  Checking $2$, we find that $2\cdot 21=42$, so our list becomes $$1\quad 2 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 21 \quad 42.$$  Checking $3$, we find that $3\cdot 14=42$, so our list becomes $$1\quad 2 \quad 3 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 14 \quad 21 \quad 42.$$  Checking $4$ and $5$ we find that $4$ and $5$ are not divisors of $42$.  Checking $6$, we find that $6\cdot 7=42$, so our list becomes $$1\quad 2 \quad 3 \quad 6 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 7 \quad 14 \quad 21 \quad 42.$$  Since $7$ is already on our list, our final list is $$1\quad 2 \quad 3 \quad 6 \quad 7 \quad 14 \quad 21 \quad 42.$$  We compare our lists for the factors of $14$ and the factors of $42$ to see that the factors that $14$ and $42$ share are $1$, $2$, $7$, and $14$.  Therefore, Rick and Steve could be thinking of $\boxed{4}$ possible numbers.  Note that since $14$ is a factor of $42$, all of the factors of $14$ are also factors of $42$.
Question:
Rick is thinking of a positive factor of $14$ and Steve is thinking of a positive factor of $42$.  If Rick and Steve are thinking of the same number, how many possible numbers could they be thinking of?
Answer:
Let's think step by step.

user
4
Rick is thinking of a positive factor of $14$ and Steve is thinking of a positive factor of $42$.  If Rick and Steve are thinking of the same number, how many possible numbers could they be thinking of?
test