The product of a pair of 2 -digit positive integers is 630 . How many such pairs are there?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
We begin by listing, in order, the smallest 2-digit positive divisors of 630 .

We note that by first writing 630 as a product of its prime factors $\left(630=2 \times 3^{2} \times 5 \times 7\right)$, it may be easier to determine these divisors.

The smallest five 2-digit positive divisors of 630 are 10,14,15,18, and 21 .

The next largest 2-digit positive divisor of 630 is 30 , and we notice that $21 \times 30=630$.

That is, 21 and 30 are a pair of 2-digit positive integers whose product is 630, and they are consecutive in the ordered list of positive divisors.

Thus, each of 10,14,15, and 18 must be paired with a divisor of 630 that is greater than 30 .

We may check that the divisors that pair with each of $10,14,15$, and 18 is a 2-digit positive integer by dividing 630 by the smaller divisor.

That is, $\frac{630}{18}=35, \frac{630}{15}=42, \frac{630}{14}=45$, and $\frac{630}{10}=63$.

Thus, the pairs of 2-digit positive integers whose product is 630 are 21 and 30, 18 and 35, 15 and 42,14 and 45 , and 10 and 63 , and so there are 5 such pairs.