Final answer:
To find the highest common factor of 5m and 3n, we need to simplify the given expressions and find their prime factorizations. The highest common factor is the product of the common prime factors. The highest common factor of 5m and 3n is 75.
Step-by-step explanation:
In order to find the highest common factor of 5m and 3n, we first need to simplify the given expressions.
Given: M = 3^4 * 5^3 = 81 * 125 = 10125
n = 3^3 * 5^2 * 11 = 27 * 25 * 11 = 7425
Now, let's find the prime factorization of 5m and 3n:
5m = 5 * (3^4 * 5^3) = 5 * 10125 = 50625
3n = 3 * (3^3 * 5^2 * 11) = 3 * 7425 = 22275
The highest common factor (HCF) is the product of the common prime factors of 50625 and 22275.
Prime factorization of 50625:
50625 = 3 * 5^4 * 13
Prime factorization of 22275:
22275 = 3 * 5^2 * 11^2
The common prime factors are 3 and 5^2.
Therefore, the highest common factor of 5m and 3n is 3 * 5^2 = 75.