SOLUTION: The number 632 is a multiple number since the first digit 6 is the product of the other two digits of the number. How many three-digit multiple numbers less than 700 exist?

1.  If the first digit is 1, there is only one way to factor 1 using two
    integers, 1×1 = 1, so the only multiple number with 1st digit 1 is 111. 

2.  If the first digit is 2, there are two ways to factor 2 using two
    integers, 1×2 = 2, and 2×1 = 2  so there are two multiple numbers with 1st
    digit 2, 212 and 221.

3.  If the first digit is 3, there are two ways to factor 3 using two
    integers, 1×3 = 3, and 3×1 = 3  so there are two multiple numbers with 1st
    digit 3, 313 and 331.

4.  If the first digit is 4, there are three ways to factor 4 using two
    integers, 1×4 = 4, 2×2 = 4 and 4×1 = 4  so there are two multiple numbers
    with 1st digit 3, 414, 422 and 441.

5.  If the first digit is 5, there are two ways to factor 5 using two
    integers, 1×5 = 5, and 5×1 = 5  so there are two multiple numbers with 1st
    digit 5, 515 and 551.

6.  If the first digit is 6, there are four ways to factor 6 using two
    integers, 1×6 = 6, 2×3 = 6, 3×2 = 6, and 661, so there are four multiple
    numbers with 1st digit 6, 616, 623, 632, and 661.

So there are 14 of them:

111, 212, 221, 313, 331, 414, 422, 441, 515, 551, 616, 623, 632, 661

Edwin