what is the GCF of 300 and 225 using prime factorization
75
To find the Greatest Common Factor (GCF) of 300 and 225 using prime factorization, we need to factorize both numbers into their prime factors, then identify the common prime factors, and finally multiply them together to find the GCF.
Let's start by finding the prime factorization of each number:
Prime factorization of 300:
300 can be divided evenly by 2, so we divide it by 2 to get 150.
150 can be divided evenly by 2, so we divide it by 2 to get 75.
75 can be divided evenly by 3, so we divide it by 3 to get 25.
25 can be divided evenly by 5, so we divide it by 5 to get 5.
5 is a prime number, so the final expression of prime factorization for 300 is: 2 x 2 x 3 x 5 x 5 = 2^2 x 3 x 5^2.
Prime factorization of 225:
225 can be divided evenly by 3, so we divide it by 3 to get 75.
75 can be divided evenly by 3, so we divide it by 3 to get 25.
25 can be divided evenly by 5, so we divide it by 5 to get 5.
5 is a prime number, so the final expression of prime factorization for 225 is: 3 x 3 x 5 x 5 = 3^2 x 5^2.
Now we compare the prime factorizations of 300 and 225, and identify the common prime factors:
Prime factorization of 300: 2^2 x 3 x 5^2
Prime factorization of 225: 3^2 x 5^2
The common prime factors between 300 and 225 are 3 and 5, raised to the power of 2 since they both have exponent 2.
Therefore, the GCF of 300 and 225 using prime factorization is 3^2 x 5^2, which is equal to 9 x 25 = 225.