Find each of the following
(a)GCD (12, 26, 65)
(b) LCM (12,26,65)
We'll be glad to check your answers.
(a)1
(b)780
To find the GCD (Greatest Common Divisor) and LCM (Least Common Multiple) of a set of numbers, we need to follow these steps:
Step 1: Prime Factorization
Find the prime factorization of each number in the set.
(a) GCD (12, 26, 65):
Prime factorization of 12: 2 * 2 * 3
Prime factorization of 26: 2 * 13
Prime factorization of 65: 5 * 13
(b) LCM (12, 26, 65):
Prime factorization of 12: 2 * 2 * 3
Prime factorization of 26: 2 * 13
Prime factorization of 65: 5 * 13
Step 2: GCD Calculation
To find the GCD, we need to determine the common prime factors and multiply them. Take the lowest power for each common factor.
(a) GCD (12, 26, 65):
Common prime factors: 2, 13
Multiply the common prime factors: 2 * 13 = 26
Therefore, GCD (12, 26, 65) = 26
Step 3: LCM Calculation
To find the LCM, we need to determine all the prime factors and multiply them with the highest power.
(a) LCM (12, 26, 65):
All prime factors: 2, 3, 5, 13
Multiply all prime factors with the highest power: 2 * 2 * 3 * 5 * 13 = 780
Therefore, LCM (12, 26, 65) = 780
So, the answers are:
(a) GCD (12, 26, 65) = 26
(b) LCM (12, 26, 65) = 780