A number theory problem by Satyabrata Dash

The conditions given are $a,b | 100^5$ , $a,b > 0$ and $a+b = 157$ . Notice that 157 is not divisible by neither 2 or 5, which are the prime divisors of $100^5$ . Thus, if both of $a,b$ were divided by any of those primes, that would cause a contradiction with the equality $a+b = 157$ as then the left-hand side would be divisible by the prime. This means that necessarily $\gcd(a,b) = 1$ . In simpler terms, they cannot share prime divisors. Thus we may say, without loss of generality, that $a = 5^x, b = 2^y$ and we now want to solve the exponential equation $5^x + 2^y = 157$ . This is easy because the powers of $5$ grow quickly:

$157 - 5^0 = 156 \\ 157 - 5 = 152 \\ 157 - 5^2 = 132 \\ 157 - 5^3 = 32 = 2^5$ . Thus $a = 125, b = 32$ . We may also reverse the order of the solution.

157=125+32