Spencer Jaffe
secondary level
asked by student

What two numbers that don't end in zero, multiplied equal 10,000,000.

Strange though it may be, this question is about the prime factorization of 10 000 000:

10 000 = 2 * 5 * 2 * 5 * 2 * 5 * 2 * 5 * 2 * 5 * 2 * 5 * 2 * 5

Now to write 10 000 000 as a product of two numbers, you just split these factors 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5 into two groups.

For instance:
first number: 2 * 2 * 2 * 5 * 2 * 2 * 5 = 800
second number: 5 * 2 * 5 * 5 * 2 * 5 * 5 = 1250.
The product of 40 and 250 is 10 000 000.

The problem is that 800 and 1250 contain zeros. Whenever a 2 and a 5 appear as factors of the same number, then this number ends in a zero. How can you split up the factors so that 2 and 5 do not appear together?