if a=2^3*3^7*5^3*11^4 and b=2^2*3^5*7^2*11*13,find the following.(leave your anwer written with exponents)a.GCD b.LCM

To find the GCD (Greatest Common Divisor) and LCM (Least Common Multiple) of two numbers, we first need to factorize the numbers into their prime factors. Given that:

a = 2^3 * 3^7 * 5^3 * 11^4
b = 2^2 * 3^5 * 7^2 * 11 * 13

a. GCD (Greatest Common Divisor):
To find the GCD, we need to determine the highest power of each common prime factor in both numbers.

In this case, the common prime factors are 2, 3, and 11.

The highest power of 2 in a is 2^3, and in b is 2^2. Therefore, the highest power of 2 that divides both numbers is 2^2.

The highest power of 3 in a is 3^7, and in b is 3^5. Therefore, the highest power of 3 that divides both numbers is 3^5.

The highest power of 11 in a is 11^4, and in b is 11^1. Therefore, the highest power of 11 that divides both numbers is 11^1.

So, the GCD of a and b is 2^2 * 3^5 * 11^1 = 132,660.

b. LCM (Least Common Multiple):
To find the LCM, we need to determine the highest power of each prime factor in either number.

In this case, the prime factors in a are 2, 3, 5, and 11, while the prime factors in b are 2, 3, 7, 11, and 13.

The highest power of 2 in a is 2^3, and in b is 2^2. Therefore, the highest power of 2 is 2^3.

The highest power of 3 in a is 3^7, and in b is 3^5. Therefore, the highest power of 3 is 3^7.

The highest power of 5 in a is 5^3, and in b is 5^1. Therefore, the highest power of 5 is 5^3.

The highest power of 7 in a is 7^0 (not present), and in b is 7^2. Therefore, the highest power of 7 is 7^2.

The highest power of 11 in a is 11^4, and in b is 11^1. Therefore, the highest power of 11 is 11^4.

The highest power of 13 in a is 13^0 (not present), and in b is 13^1. Therefore, the highest power of 13 is 13^1.

So, the LCM of a and b is 2^3 * 3^7 * 5^3 * 7^2 * 11^4 * 13^1 = 1,233,854,000.

Therefore:
a. GCD = 132,660
b. LCM = 1,233,854,000