I’m thinking of two positive integers. When they are multiplied together, their product is one billion. Neither of these integers ends in a zero. What is the sum of these two numbers?

Solution

First let’s work out the prime factors of a billion.

1,000,000,000 = 2^{9} × 5^{9}

The only way to get a zero at end of a number is if that number is multiple of 10, which is created by the product of the prime factors 2 and 5. Since neither number ends in zero, all the factors of two must be in one of the numbers, and all the factors of five must be in the other number.

N_{1} = 2^{9} = 512
N_{2} = 5^{9} = 1,953,125

The answer is that the sum of the two numbers is 1,953,637

N_{1} + N_{2} = 1,953,637

Have you ever wondered how long it would take you to count to a billion? How about a trillion?