Start with the prime factorization of 1,000,000
= 2 * 2 * 2 * 2 * 2 * 2 * 5 * 5 * 5 * 5 * 5 * 5.
So the two integers must consist of six 2s and six 5s between the two of them.
Any integer that contains a 2 and 5 in it’s prime factorization will end in 0. (2 * 5 = 10 and any integer multiplied by 10 ends in 0.)
=> one integer contains all six 2s and the other one contains all six 5s.
I read this riddle when I didn’t have a calculator nearby (it must have been on shabbos) and the way I calculated (2^6) and (5^6) is by breaking it into squares.
(2^6) = (2^3)^2, we all know that 2^3 = 8 and 8^2 = 64.
(5^6) = (5^3)^2 was a little trickier. 5 * 5 = 25 and 25 * 5 = 125 (5 quarters is 125 cents right?) now 125^2 is what?
Using the trick from the second half of this post http://www.theyeshivaworld.com/coffeeroom/topic/the-riddle-thread/page/8#post-16043
125 * 125 = 100 * ( 12 * 13) + 25 = 15,625.
The two integers are 64 and 15,625 and that is the only solution.