Q: What is the prime factorization of the number 761,737,536?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7 x 137 x 197
    • or also written as { 2, 2, 2, 2, 2, 2, 3, 3, 7, 7, 137, 197 }
  • Written in exponential form: 26 x 32 x 72 x 1371 x 1971

Why is the prime factorization of 761,737,536 written as 26 x 32 x 72 x 1371 x 1971?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 761,737,536

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 761,737,536 by 2

761,737,536 ÷ 2 = 380,868,768 - No remainder! 2 is one of the factors!
380,868,768 ÷ 2 = 190,434,384 - No remainder! 2 is one of the factors!
190,434,384 ÷ 2 = 95,217,192 - No remainder! 2 is one of the factors!
95,217,192 ÷ 2 = 47,608,596 - No remainder! 2 is one of the factors!
47,608,596 ÷ 2 = 23,804,298 - No remainder! 2 is one of the factors!
23,804,298 ÷ 2 = 11,902,149 - No remainder! 2 is one of the factors!
11,902,149 ÷ 2 = 5,951,074.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
11,902,149 ÷ 3 = 3,967,383 - No remainder! 3 is one of the factors!
3,967,383 ÷ 3 = 1,322,461 - No remainder! 3 is one of the factors!
1,322,461 ÷ 3 = 440,820.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
1,322,461 ÷ 5 = 264,492.2 - This has a remainder. 5 is not a factor.
1,322,461 ÷ 7 = 188,923 - No remainder! 7 is one of the factors!
188,923 ÷ 7 = 26,989 - No remainder! 7 is one of the factors!
26,989 ÷ 7 = 3,855.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
26,989 ÷ 11 = 2,453.5455 - This has a remainder. 11 is not a factor.
26,989 ÷ 13 = 2,076.0769 - This has a remainder. 13 is not a factor.
26,989 ÷ 17 = 1,587.5882 - This has a remainder. 17 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
26,989 ÷ 137 = 197 - No remainder! 137 is one of the factors!
197 ÷ 137 = 1.438 - There is a remainder. We can't divide by 137 evenly anymore. Let's try the next prime number
197 ÷ 139 = 1.4173 - This has a remainder. 139 is not a factor.
197 ÷ 149 = 1.3221 - This has a remainder. 149 is not a factor.
197 ÷ 151 = 1.3046 - This has a remainder. 151 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
197 ÷ 197 = 1 - No remainder! 197 is one of the factors!

The orange divisor(s) above are the prime factors of the number 761,737,536. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7 x 137 x 197 = 761,737,536. It can also be written in exponential form as 26 x 32 x 72 x 1371 x 1971.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 761,737,536.

761,737,536
Factor Arrows
2380,868,768
Factor Arrows
2190,434,384
Factor Arrows
295,217,192
Factor Arrows
247,608,596
Factor Arrows
223,804,298
Factor Arrows
211,902,149
Factor Arrows
33,967,383
Factor Arrows
31,322,461
Factor Arrows
7188,923
Factor Arrows
726,989
Factor Arrows
137197

More Prime Factorization Examples

761,737,534761,737,535761,737,537761,737,538
21 x 12,7131 x 29,959151 x 131 x 2771 x 42,3071111 x 371 x 1,871,591121 x 380,868,7691

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