Question

find m and n so that the prime factorization of 10500 is expressed as 2^m*3*5^n*7

Answer 1

10500 = 3 × 3500
10500 = 3 × 2 × 1750
10500 = 3 × 2 × 2 × 875
10500 = 3 × 2 × 2 × 5 × 175
10500 = 3 × 2 × 2 × 5 × 5 × 35 
10500 = 2 × 2 × 3 × 5 × 5 × 5 × 7

Here we see that 2³ × 3 × 5³ × 7 = 2 × 2 × 3 × 5 × 5 × 5 × 7
Now it's simple.

The first factor is 2, and in the equation 2 is raised to m. There are two 2s in the factorization. Therefore m=2. 
Then 3 is present in both. 5 is then raised to n. 5 comes 3 times. Therefore n=3.
Then the 7 appears in both equations.

Ans: m=2 and n=3