How do you find the prime factorizatoin of a number?

Answer 1

Prime Factorization

"Prime Factorization" is finding which prime numbers you need to multiply together to get the original number.

Example 1

What are the prime factors of 12?

It is best to start working from the smallest Prime number, which is 2, so let's check:

12 ÷ 2 = 6

But 6 is not a prime number, so we need to factor it further:

6 ÷ 2 = 3

And 3 is a prime number, so:

12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right - the prime factorization of 12 is 2 × 2 × 3, which can also be written as 22 × 3

Example 2

What is the prime factorization of 147?

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3:

147 ÷ 3 = 49

Then we try factoring 49, and find that 7 is the smallest prime number that works:

49 ÷ 7 = 7

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7 = 3 × 72

Answer 2

Technically, you can't. You can find the prime factors of a number. If you compare those prime factors to the prime factors of another number, you will see if they have any prime factors in common.

Answer 3

To find the prime factorization, try to divide the number by any prime number that is less than the square root of the number (one of any factor pair must be equal to or smaller than the square root). The number is expressed as a product of a string of primes.

Example: 850 = 2 x 425 = 2 x 5 x 85 = 2 x 5 x 5 x 17 (all primes)

Attempt division of the number by 2, 3, 5, 7, 11, 13, 17, 23 and so forth, and create a list of any multiple occurrences. There are some shortcuts you can use:

- if a number is even, divide by 2 as many times as necessary to get an odd number result

- if a number ends in 5, immediately divide by 5 as many times as necessary to get a result that does not end in 5

- if the individual digits of the number add up to 9 (e.g. 27, 72, 108) or a multiple of 9 (882 = 18, 9828 = 27), then the number is divisible by 9 (3 x 3)

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Extended Example: 1962

First, set up the initial equation:

1962 = 1962

Then, try to divide the biggest number on the right side of the equation each time by the smallest prime not already considered.

1962 = 2 * 981

In the example, considering 981 / 2, it doesn't work this time, so move on to 3.

1962 = 2 * 3 * 327

327 is also divisible by 3, so we should repeat the division.

1962 = 2 * 3 * 3 * 109

Since 109 is below 121, you only need to check if it's divisible by 3, 5, and 7 to see that it's prime. So, in the example, the last line is the full prime factorization for 1962.

If you want to express it in exponential form, just count how many times each number appears in the full prime factorization, and give the number that exponent.

1962 = 21 * 32 * 1091

Shortcut Example: 1962

As explained above, it's a number whose digits add up to 9 or a multiple of 9.

1962 = 9 x 218 = 3 x 3 x 218 = 2 x 3 x 3 x 109

Prime Factorization

"Prime Factorization" is finding which prime numbers you need to multiply together to get the original number.

Example 1

What are the prime factors of 12?

It is best to start working from the smallest prime number, which is 2, so let's check:

12 ÷ 2 = 6

But 6 is not a prime number, so we need to factor it further:

6 ÷ 2 = 3

And 3 is a prime number, so:

12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right - the prime factorization of 12 is 2 × 2 × 3, which can also be written as 22 × 3

Example 2

What is the prime factorization of 147?

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3:

147 ÷ 3 = 49

Then we try factoring 49, and find that 7 is the smallest prime number that works:

49 ÷ 7 = 7

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7 = 3 × 72
All composite numbers can be expressed as unique products of prime numbers. This is accomplished by dividing the original number and its factors by prime numbers until all the factors are prime. A factor tree can help you visualize this.

Example: 210

210 Divide by two.

105,2 Divide by three.

35,3,2 Divide by five.

7,5,3,2 Stop. All the factors are prime.

2 x 3 x 5 x 7 = 210

That's the prime factorization of 210.

Answer 4

All composite numbers can be expressed as unique products of prime numbers. This is accomplished by dividing the original number and its factors by prime numbers until all the factors are prime. A factor tree can help you visualize this.
Example: 210

210 Divide by two.
105,2 Divide by three.
35,3,2 Divide by five.
7,5,3,2 Stop. All the factors are prime.

2 x 3 x 5 x 7 = 210
That's the prime factorization of 210.

Answer 5

All composite numbers can be expressed as unique products of prime numbers. This is accomplished by dividing the original number and its factors by prime numbers until all the factors are prime. A factor tree can help you visualize this.

Example: 210

210 Divide by two.

105,2 Divide by three.

35,3,2 Divide by five.

7,5,3,2 Stop. All the factors are prime.

2 x 3 x 5 x 7 = 210

That's the prime factorization of 210.