Find The Prime Factorization Of 63 Using Exponents - CoolConversion

What is prime factorization?

Definition of prime factorization

The prime factorization is the decomposition of a composite number into a product of prime factors that, if multiplied, recreate the original number. Factors by definition are the numbers that multiply to create another number. A prime number is an integer greater than one which is divided only by one and by itself. For example, the only divisors of 7 are 1 and 7, so 7 is a prime number, while the number 72 has divisors deived from 23×32 like 2, 3, 4, 6, 8, 12, 24 ... and 72 itself, making 72 not a prime number. Note the the only "prime" factors of 72 are 2 and 3 which are prime numbers.

Prime factorization example 1

Let's find the prime factorization of 72.

Solution 1

Start with the smallest prime number that divides into 72, in this case 2. We can write 72 as: 72 = 2 × 36 Now find the smallest prime number that divides into 36. Again we can use 2, and write the 36 as 2 × 18, to give. 72 = 2 × 2 × 18 18 also divides by 2 (18 = 2 × 9), so we have: 72 = 2 × 2 × 2 × 9 9 divides by 3 (9 = 3 × 3), so we have: 72 = 2 × 2 × 2 × 3 × 3 2, 2, 2, 3 and 3 are all prime numbers, so we have our answer. In short, we would write the solution as: 72 = 2 × 36 72 = 2 × 2 × 18 72 = 2 × 2 × 2 × 9 72 = 2 × 2 × 2 × 3 × 3 72 = 23 × 32 (prime factorization exponential form)

Solution 2

Using a factor tree:

• Procedure:
• Find 2 factors of the number;
• Look at the 2 factors and determine if at least one of them is not prime;
• If it is not a prime factor it;
• Repeat this process until all factors are prime.

See how to factor the number 72:

72 / \ 2 36 / \ 2 18 / \ 2 9 / \ 3 3

72 is not prime --> divide by 2 36 is not prime --> divide by 2 18 is not prime --> divide by 2 9 is not prime --> divide by 3 3 and 3 are prime --> stop

Taking the left-hand numbers and the right-most number of the last row (dividers) an multiplying then, we have

72 = 2 × 2 × 2 × 3 × 3

72 = 23 × 32 (prime factorization exponential form)

Note that these dividers are the prime factors. They are also called the leaves of the factor tree.

Prime factorization example 2

See how to factor the number 588:

588 / \ 2 294 / \ 2 147 / \ 3 49 / \ 7 7

588 is not prime --> divide by 2 294 is not prime --> divide by 2 147 is not prime --> divide by 3 49 is not prime --> divide by 7 7 and 7 are prime --> stop

Taking the left-hand numbers and the right-most number of the last row (dividers) an multiplying then, we have

588 = 2 × 2 × 3 × 7 × 7 588 = 22 × 3 × 72 (prime factorization exponential form)