Learn how to find the GCF of 30 and 54 using prime factorization and division methods. Solve GCF problems effectively with step-by-step techniques.
Finding the GCF of 30 and 54
Prime Factorization Method
To find the (GCF) of 30 and 54 using the prime factorization method, we need to determine the prime factors of both numbers and then identify the common factors.
First, let’s factorize 30. We start by dividing it by the smallest prime number, which is 2. Since 30 is divisible by 2, we divide it by 2 to get 15. Now, we continue dividing by 2 until we can no longer divide evenly. Thus, the prime factorization of 30 is 2 × 3 × 5.
Next, let’s factorize 54. Again, we begin by dividing it by 2. Since 54 is divisible by 2, we divide it by 2 to obtain 27. Continuing with the process, we divide 27 by 3, resulting in 9. Finally, we divide 9 by 3 to get 3. Hence, the prime factorization of 54 is 2 × 3 × 3 × 3.
Now that we have the prime factorizations of both numbers, we can identify the common factors. Looking at the factors of 30 (2, 3, and 5) and the factors of 54 (2 and 3), we find that the only common factors are 2 and 3. Multiplying these common factors together, we get 2 × 3 = 6. Therefore, the GCF of 30 and 54 using the prime factorization method is 6.
Division Method
Another approach to finding the GCF of 30 and 54 is the division method. This method involves dividing the two numbers and repeatedly dividing the remainder by the previous divisor until the remainder becomes zero.
To begin, we divide 54 by 30, which gives us a quotient of 1 and a remainder of 24. Next, we divide 30 by 24, resulting in a quotient of 1 and a remainder of 6. Continuing the process, we divide 24 by 6, giving us a quotient of 4 and no remainder.
At this point, since the remainder is zero, we stop dividing. The last divisor used, which is 6 in this case, is the GCF of 30 and 54. Therefore, using the division method, we find that the GCF of 30 and 54 is 6.
In summary, both the prime factorization method and the division method lead us to the same result, which is a GCF of 6 for the numbers 30 and 54.
