How do you find the LCM and the GCF for 72x and 150y?
72x = 2 * 2 * 2 * 3 * 3 * x
150y = 2 * 3 * 5 * 5 * y
Can you work it from there?
Thanks
To find the Least Common Multiple (LCM) and the Greatest Common Factor (GCF) for 72x and 150y, we will break down each term into its prime factors.
Step 1: Prime factorization of 72x
To factorize 72, we divide it by the smallest prime number, which is 2:
72 ÷ 2 = 36
Next, we divide 36 by 2:
36 ÷ 2 = 18
Again, divide 18 by 2:
18 ÷ 2 = 9
Since 9 is already a prime number, we stop. So, the prime factorization of 72 is 2 * 2 * 2 * 3 * 3 = 2^3 * 3^2
Now, we look at the variable x. Since there are no other factors, we can say that x = x^1
So, the prime factorization of 72x is 2^3 * 3^2 * x
Step 2: Prime factorization of 150y
To factorize 150, we start with 2:
150 ÷ 2 = 75
Next, divide 75 by 3:
75 ÷ 3 = 25
Since 25 is already a prime number, we stop. So, the prime factorization of 150 is 2 * 3 * 5^2 = 2^1 * 3^1 * 5^2
Now, we look at the variable y. Since there are no other factors, we can say that y = y^1
So, the prime factorization of 150y is 2^1 * 3^1 * 5^2 * y
Step 3: Finding the LCM
To find the LCM, we take the highest power of each prime factor that appears in either prime factorization.
LCM = 2^3 * 3^2 * 5^2 * x * y = 8 * 9 * 25 * xy = 1800xy
So, the LCM of 72x and 150y is 1800xy.
Step 4: Finding the GCF
To find the GCF, we take the lowest power of each prime factor that appears in both prime factorizations.
GCF = 2^1 * 3^1 = 2 * 3 = 6
So, the GCF of 72x and 150y is 6.