How many different pairs of positive whole numbers have a greatest common factor of 4 and a lowest common multiple of 4620 ?
(A) 4
(B) 5
(C) 7
(D) 8
(E) 11
We begin by considering the prime factors (called the prime factorization) of each of the two numbers, 4 and 4620 .

$$
\begin{gathered}
4=2 \times 2 \\
4620=2 \times 2 \times 3 \times 5 \times 7 \times 11
\end{gathered}
$$

Let a pair of positive whole numbers whose greatest common factor is 4 and whose lowest common multiple is 4620 be $(a, b)$.

Since the pair $(a, b)$ has a greatest common factor of 4 , then each of $a$ and $b$ is a multiple of 4 and thus $2 \times 2$ is included in the prime factorization of each of $a$ and $b$.

Further, each of $a$ and $b$ cannot have any other prime factors in common otherwise their greatest common factor would be greater than 4 .

The lowest common multiple of $a$ and $b$ is 4620 and so each of $a$ and $b$ is less than or equal to 4620 .

Further, if for example $a$ has prime factors which are not prime factors of 4620 then 4620 is not a multiple of $a$.

That is, each of $a$ and $b$ must have prime factors which are chosen from 2, 2, 3, 5, 7 , and 11 only. Summarizing, $a$ is a positive whole number of the form $2 \times 2 \times m$ and $b$ is a positive whole number of the form $2 \times 2 \times n$, where:

- $m$ and $n$ are positive whole numbers
- the prime factors of $m$ are chosen from 3, 5, 7, 11 only
- the prime factors of $n$ are chosen from 3,5,7,11 only
- $m$ and $n$ have no prime factors in common

In the table below, we list the possible values for $m$ and $n$ which give the possible pairs $a$ and $b$. To ensure that we don't double count the pairs $(a, b)$, we assume that $a \leq b$ and thus $m \leq n$.

| $m$ | $n$ | $a$ | $b$ | $(a, b)$ |
| :---: | :---: | :---: | :---: | :---: |
| 1 | $3 \times 5 \times 7 \times 11$ | $2 \times 2 \times 1$ | $2 \times 2 \times 3 \times 5 \times 7 \times 11$ | $(4,4620)$ |
| 3 | $5 \times 7 \times 11$ | $2 \times 2 \times 3$ | $2 \times 2 \times 5 \times 7 \times 11$ | $(12,1540)$ |
| 5 | $3 \times 7 \times 11$ | $2 \times 2 \times 5$ | $2 \times 2 \times 3 \times 7 \times 11$ | $(20,924)$ |
| 7 | $3 \times 5 \times 11$ | $2 \times 2 \times 7$ | $2 \times 2 \times 3 \times 5 \times 11$ | $(28,660)$ |
| 11 | $3 \times 5 \times 7$ | $2 \times 2 \times 11$ | $2 \times 2 \times 3 \times 5 \times 7$ | $(44,420)$ |
| $3 \times 5$ | $7 \times 11$ | $2 \times 2 \times 3 \times 5$ | $2 \times 2 \times 7 \times 11$ | $(60,308)$ |
| $3 \times 7$ | $5 \times 11$ | $2 \times 2 \times 3 \times 7$ | $2 \times 2 \times 5 \times 11$ | $(84,220)$ |
| $3 \times 11$ | $5 \times 7$ | $2 \times 2 \times 3 \times 11$ | $2 \times 2 \times 5 \times 7$ | $(132,140)$ |

There are 8 different pairs of positive whole numbers having a greatest common factor of 4 and a lowest common multiple of 4620 .

ANswer: (D)