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 need to factorize them into their prime factors and then compare the exponents.
Given:
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 of two numbers, we take the lowest exponent for each prime factor that both numbers share.
Prime Factors:
- The numbers share the prime factors 2, 3, and 11.
Exponents:
- For 2, the lowest exponent present in both numbers is 2^2.
- For 3, the lowest exponent present in both numbers is 3^5.
- For 11, the lowest exponent present in both numbers is 11.
Therefore, the GCD of a and b is:
GCD(a, b) = 2^2 * 3^5 * 11 = 4 * 243 * 11 = 106884.
b. LCM (Least Common Multiple):
To find the LCM of two numbers, we take the highest exponent for each prime factor found in both numbers.
Prime Factors:
- The numbers have the prime factors 2, 3, 5, 7, 11, and 13.
Exponents:
- For 2, the highest exponent present in both numbers is 2^3.
- For 3, the highest exponent present in both numbers is 3^7.
- For 5, the exponent in a is 5^3 and in b, it is not present, so we take the highest exponent from a, i.e., 5^3.
- For 7, the exponent in a is not present, and in b, it is 7^2, so we take the highest exponent from b, i.e., 7^2.
- For 11, the highest exponent present in both numbers is 11^4.
- For 13, the exponent in a is not present, and in b, it is 13, so we take the highest exponent from b, i.e., 13.
Therefore, the LCM of a and b is:
LCM(a, b) = 2^3 * 3^7 * 5^3 * 7^2 * 11^4 * 13 = 864,360,600.