The license plate on my car has a three-digit number on it.
The product of the digits of this three-digit number is
216,
and their sum is 19 and the digits appear in ascending order.
Find the license plate number.
-
You can guess at this one, but it will take a lot of guessing!
- To be more systematic about it, let's think about the divisors of
216. There are 16 of them: 1, 2, 3, 4, 6, 8, 9 and nine more that
are larger than 10.
-
So the digits of our licence plate have to be chosen from the ones
listed above.
-
Think a little more about this: the prime factorization of
216 is (23)(33) -- so we need three factors
of 3. Since only 3, 6 and 9 have factors of three, we see that one of
the digits must be 9 (since otherwise we can have at most two factors
of 3 in the product).
-
Once we've settled on the 9, there are only three ways to pick the
other
digits to add up to 10 (so that together all three digits will add up to
19):
1 and 9, 2 and 8, and 4 and 6.
-
Now you can just try the three possibilities: 199 gives a product of
81, so that's not
it. 288 gives a product of 128, so that's not it. But 469 gives a product
of 216.
-
So the license number was 469.
