The prime factorization of 72 is 23×322 cubed cross 3 squared 23×32 .
Understanding Prime Factorization
Prime factorization is the process of breaking down a composite number into its prime number components. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (examples include 2, 3, 5, 7, 11, etc.). A composite number is a whole number greater than 1 that has more than two divisors. The fundamental theorem of arithmetic states that every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers, and this representation is unique.
The Factorization Process for 72
To find the prime factorization of 72, we can systematically divide the number by the smallest prime numbers until we are left with a prime number.
Step 1: Divide by the Smallest Prime Number
We begin by dividing 72 by the smallest prime number, which is 2.
72÷2=3672 divided by 2 equals 36
72÷2=36
Step 2: Continue Dividing the Result
The result, 36, is an even number, so it is also divisible by 2.
36÷2=1836 divided by 2 equals 18
36÷2=18
The new result, 18, is still an even number, so we divide by 2 again.
18÷2=918 divided by 2 equals 9
18÷2=9
Step 3: Move to the Next Prime Number
The result, 9, is no longer divisible by 2. We move to the next prime number, which is 3.
9÷3=39 divided by 3 equals 3
9÷3=3
Step 4: The Final Prime Factor
The result is 3, which is a prime number. We can stop here, as we have successfully broken down 72 into its prime factors.
Assembling the Prime Factors
Now, we collect all the prime numbers we used in the division process: 2, 2, 2, 3, and 3.
The prime factors of 72 are 2×2×2×3×32 cross 2 cross 2 cross 3 cross 3
2×2×2×3×3
.
Expressing in Exponential Form
To express this as a product of the powers of the prime factors, we count how many times each prime number appears.
-
The number 2 appears three times, so we write it as 232 cubed
23
.
-
The number 3 appears two times, so we write it as 323 squared
32
.
Combining these, we get the final prime factorization: **23×322 cubed cross 3 squared
23×32** . This shows that the number 72 is composed of the product of three 2s and two 3s.
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