### Instruction: What is the product of the positive integer divisors of 100?### Response: I know that 100 has the prime factorization 2^2 * 5^2, so its positive integer divisors are all the products of the form 2^a * 5^b, where a and b are 0, 1, or 2.
There are nine such divisors: 1, 2, 4, 5, 10, 20, 25, 50, and 100.
I could multiply them all together, but that would be tedious and prone to error.
Instead, I notice that there is a symmetry in the divisors: for every divisor d, 100/d is also a divisor.
So I can pair up the divisors as follows: 1 * 100, 2 * 50, 4 * 25, 5 * 20, and 10 * 10.
The product of each pair is 100, so the product of all the divisors is 100^5.