No 0's at all!

Note that $10000 = 10^4 = 2^4 \cdot 5^4.$ Any number which has 2 AND 5 as factors will end in a 0, so the only possible way to write 10000 as a product of two numbers which don't contain 0s is $10000 = 2^4 \cdot 5^4 = 16 \cdot 625$ , so the sum is $16+625 = 641.$

Without writing down, this is pretty much random guess from well known factors

10,000 should = xx5 * xx2

And, it's known that 125 * 8 = 1,000 => 10,000 = 125 * 80. To eliminate zero from 80, divide it by something odd, like 5: 80 / 5 = 16. Next: 125 * 5 = 625

=> 10,000 = 16 * 625

-- Edited: Just read other comments, I think prime factorization is the formal academic solution to this.

just refer to Fox' theorem

10000 = 100x100= 25x4x25×4 = 625*16. Hence the required sum is 625+16 = 641

A×B=100×100=(4×25)×(4×25)=16×625,A+B=641##

It can be easily solved by prime factorization method. Prime factors of 10000 can be written as Now separate the terms which do not contain 0 as digit. The only way to do this is to keep all 2s and 5s separate i.e. And hence,

10000=2^4*5^4=10^4 2^4=16 5^4=625 16+625=641. Both 625 and 16 have no 0s.

At first glance, we all think that 10000 is 10x10x10x10(correct), but the secret here is that 10=2x5

So 10000 =10x10x10x10=2x2x2x2x5x5x5x5=2^4 x 5^4.

2^4=16

5^4=625

625+16=641

I know these sums are messy. Sorry!

Expressed as a product of its prime factors, $10000=2^4 \cdot 5^4$

$A=2^4=16$

$B=5^4=625$

$\therefore A+B=16+625=641$

625 x 16 = 10000, 625 + 16 = 641

Factors 10000 are four 2's and four 5's. We get a zero only when we multiply 2&5 . so answer would be 5^4+2^4