The prime factorization of the greatest common factor (GCF) of 2 4 and 28 is 222 squared 22 , which equals 4. This result is found by first determining the prime factorization of each number and then finding the common prime factors.
Step 1: Prime factorization of 24
To find the prime factors of 24, divide it by the smallest prime number possible and continue until the result is 1.
-
24÷2=1224 divided by 2 equals 12
24÷2=12
-
12÷2=612 divided by 2 equals 6
12÷2=6
-
6÷2=36 divided by 2 equals 3
6÷2=3
-
3÷3=13 divided by 3 equals 1
3÷3=1
The prime factorization of 24 is 2×2×2×32 cross 2 cross 2 cross 3
2×2×2×3
, or 23×312 cubed cross 3 to the first power
23×31
.
Step 2: Prime factorization of 28
Similarly, find the prime factors of 28.
-
28÷2=1428 divided by 2 equals 14
28÷2=14
-
14÷2=714 divided by 2 equals 7
14÷2=7
-
7÷7=17 divided by 7 equals 1
7÷7=1
The prime factorization of 28 is 2×2×72 cross 2 cross 7
2×2×7
, or 22×712 squared cross 7 to the first power
22×71
.
Step 3: Find the GCF
The GCF is the product of the lowest powers of the common prime factors. In this case, the only common prime factor is 2.
-
The lowest power of 2 in the factorizations is 222 squared
22
(from 28), since 232 cubed
23
is a higher power.
-
The prime number 3 is a factor of 24 but not 28, so it is not a common factor.
-
The prime number 7 is a factor of 28 but not 24, so it is not a common factor.
Therefore, the prime factorization of the GCF is 222 squared
22
.
Step 4: Calculate the GCF
Finally, calculate the value of the GCF from its prime factorization.
-
GCF=22=2×2=4cap G cap C cap F equals 2 squared equals 2 cross 2 equals 4
𝐺𝐶𝐹=22=2×2=4
This means that 4 is the largest number that can divide both 24 and 28 without leaving a remainder.