Question

Two numbers with no zeros in their make-up can be multiplied to create 10,000. They are
625 and 16. Is it possible to have two numbers multiplied together with no zeros that equal
100,000? 1,000,000?

Answer 1

Answer:

Others have given the correct answer, but I’ll try to explain the intuition for how you would solve this.

Our number n is 10^9, so its prime factorization is 2^9 * 5^9.

Now choosing two numbers that multiply to n is equivalent to dividing these prime factors into two sets. For example, I could have the first set contain five 2’s and four 5’s, and the other contain four 2’s and five 5’s. Then my two factors would be a = 2^5 * 5^4, b = 2^4 * 5^5.

Note that a number ends in zero when it is divisible by 10. This happens when it has at least one factor of 2 and at least one factor of 5.

So the only solution is to divide our primes into two sets, in which one contains all of the 2’s, and the other contains all of the 5’s. Our two factors are then 2^9 = 512, and 5^9 = 1,953,125.

Step-by-step explanation: