Find the product of the proper divisors of 720

Sure, let's find the product of the proper divisors of 720 step by step.

Step 1: Find the prime factorization of 720. 720 = 2^4 * 3^2 * 5

Step 2: Find all the divisors of 720. The divisors of 720 are the combinations of its prime factors. They are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720.

Step 3: Exclude the number itself (720) to get the proper divisors. The proper divisors of 720 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360.

Step 4: Find the product of the proper divisors. The product of the proper divisors of 720 is the product of all these numbers. However, calculating this directly would result in a very large number.

A more efficient way to calculate this is to realize that every divisor d of 720 pairs with a divisor 720/d to multiply to 720. Therefore, the product of all divisors of 720 is 720^(number of divisors / 2).

The number of divisors of 720 is 30 (including 720 itself), so the number of proper divisors is 29. Therefore, the product of the proper divisors of 720 is 720^(29/2), which is a very large number.