Question

Compute without a calculator:

$$(72\cdot 78\cdot 85\cdot 90\cdot 98)\div (68\cdot 84\cdot 91\cdot 108).$$
(There's an easier way than multiplying out the giant products $72\cdot 78\cdot 85\cdot 90\cdot 98$ and $68\cdot 84\cdot 91\cdot 108$!)

Answer 1

Take the prime factorization of each factor in the product.

`(72 * 78 * 85 * 90 *98)/(68 * 84 * 91 * 108) \\\\ ~~~~~~~~ = ((2^3*3^2)*(2*3*13)*(5*17)*(2*3^2*5)*(2*7^2))/((2^2*17)*(2^2*3*7)*(7*13)*(2^2*3^3)) \\\\ ~~~~~~~~ = (2^6 * 3^5 * 5^2 * 7^2 * 13 * 17)/(2^6 * 3^4 * 7^2 * 13 * 17) \\\\ ~~~~~~~~ = 3 * 5^2 = \boxed{75}`