2014 AMC 10A Problems/Problem 20

The following problem is from both the 2014 AMC 12A #16 and 2014 AMC 10A #20, so both problems redirect to this page.

Problem

The product , where the second factor has digits, is an integer whose digits have a sum of . What is ?

Solution

We can list the first few numbers in the form

(Hard problem to do without the multiplication, but you can see the pattern early on)

By now it's clear that the numbers will be in the form , s, and . We want to make the numbers sum to 1000, so . Solving, we get , meaning the answer is

Solution 2(Educated Guesses if you have no time)

We first note that and so we assume there are 8s.

Then we note that it is asking for the second factor, so we subtract (the original in the first factor).

Now we have The second factor is obviously a multiple of .

Listing the first few, we have

We notice that the 4th answer choice is 1 less than a 992(a multiple of 124.)

Thus we make an educated guess that it is somehow less by 1, so we get . ~mathboy282

Note(Must Read)

We were just lucky; this method is NOT reliable. Please note that this may not work for other problems.

Video Solution

https://youtu.be/wQzuQZvq8sk

~savannahsolver

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.