The product of two whole numbers is 10,000. Neither contains a zero. What are the two numbers?

Question

Find the product of  two whole numbers is 10,000. Neither of the whole numbers has a zero.

Answer 1

Factor 10,000:

 

10000 = 10×10×10×10 = 2×5×2×5×2×5×2×5 = 2⁴×5⁴

 

 

That means we're looking for two numbers that, together, are made of exactly four 2's and four 5's. As soon as you combine a 2 and a 5 together in one of the numbers, you've got a multiple of 2×5 = 10, which definitely has a zero in it. (For example, 2×2×5 = 2×10 = 20, or 5×5×5×2 = 25×10 = 250.) So, we've got to keep the four 2's separate from the four 5's. That means your two numbers have to be:

 

2×2×2×2 = 16

5×5×5×5 = 625

 

 

16×625 = 10000

Answer 2

625*16=10000