Can you find two whole numbers, with the smallest possible difference between them, which when multipled together equal: 1234567890?

Thank you,

Bradley Kloetzly
Grade 11, Student

This is an interesting question. I first took the square root of 1 234 567 890 to see if I could guess the correct answer. The square root is 35 136.4 which didn't help.

The "right" way to solve this problem is to find the prime factors of 1 234 567 890 and then arrange the prime into two groups to give two factors whose product is 1 234 567 890. The challenge I expected to be the choice of the two groups so that the two resulting factors would have a minimum difference.

Clearly 1 234 567 890 is divisible by 10 so 2 and 5 are prime factors. Thus 1 234 567 890 = 2x5x123 456 789. 1+2+3+4+5+6+7+8+9 = 45 which is divisible by 9 so 123 456 789 is divisible by 9. The division produces

Faced with finding the prime factors of 13 717 421 I resorted to using mathematical software, Mathematica, which gave 13 717 421 as the product of the two primes 3607 and 3803. Thus the prime factorization is

The answer to the original problem is then clear. The factors are