Composite Number 472 Prime Factorization (Decomposing, Breaking It Down Into Prime Factors), Written as a Product of Primes (With Exponents, as Powers)

The prime factorization (the decomposition into prime factors) of the composite number 472

472 is not a prime number but a composite one.

The prime factorization (the decomposition into prime factors) of the composite number 472:

~ The prime factorization written as a product of prime factors:

472 = 2 × 2 × 2 × 59

~ The prime factorization written in a condensed way - as a product of powers (at least some prime factors are raised to an exponent): *

472 = 23 × 59

* A power, or a number written with exponents, is a base raised to the exponent (we say: the base raised to the power of the exponent). The exponent indicates how many times the base is multiplied by itself: 53 = 5 × 5 × 5 = 125. We say 5 raised to the power of 3. 53 is the power, 5 is the base, 3 is the exponent and 125 is the value of the power.




[1] The prime factorization of a number (the decomposition into prime factors): finding the prime numbers that multiply together to make that number.
Example: 12 = 2 × 2 × 3 = 22 × 3.


[2] Prime number: a natural number that is divisible (it is divided without a remainder) only by 1 and itself. A prime number has only two factors: 1 and the number itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19, 23.
The first prime number is 2 and not 1. The number 1 is not considered a prime number. The only even prime number is 2. All the other prime numbers are odd numbers.

[3] Composite number: a natural number that has at least one factor other than 1 and itself. A composite number has at least three factors. A composite number is also a number that is not a prime number.
Example: 4, 6, 8, 9, 10, 12, 14, 15, 16.
The composite numbers are made of prime numbers that are multiplied together.

The numbers 0 and 1 are considered neither prime nor composite numbers.


The prime factorization of a number, how is it done?

Let's learn by having an example:

Take the number 220 and build its prime factorization


We need the list of the first prime numbers, ordered from 2 up to, let's say, 20:
2, 3, 5, 7, 11, 13, 17, 19.
The prime numbers are the building blocks of the composite numbers.


1. Start by dividing 220 by the smallest prime number, 2:
220 ÷ 2 = 110; remainder = 0 ⇒
220 is divisible by 2 ⇒ 2 is a prime factor of 220:
220 = 2 × 110.

2. Divide the result of the previous operation, 110, by 2, again:
110 ÷ 2 = 55; remainder = 0 ⇒
110 is divisible by 2 ⇒ 2 is a prime factor of 110:
220 = 2 × 110 = 2 × 2 × 55.


3. Divide the result of the previous operation, 55, by 2, again:
55 ÷ 2 = 27 + 1; remainder = 1 ⇒
55 is not divisible by 2.


4. Move on to the next prime number, 3. Divide 55 by 3:
55 ÷ 3 = 18 + 1; remainder = 1 ⇒
55 is not divisible by 3.


5. Move on to the next prime number, 5. Divide 55 by 5:
55 ÷ 5 = 11; remainder = 0 ⇒
55 is divisible by 5 ⇒ 5 is a prime factor of 55:
220 = 2 × 2 × 55 = 2 × 2 × 5 × 11.


6. Notice that the remaining factor, 11, is a prime number, so we've already found all the prime factors of 220.


Conclusion, the prime factorization of 220:
220 = 2 × 2 × 5 × 11.
This can be written in a condensed form, in exponential notation:
220 = 22 × 5 × 11.

Check whether a number is prime or not. Run the prime factorization of the composite numbers

The prime factorization of a number N = Dividing the number N into smaller numbers that are prime. By multiplying these smaller prime numbers one gets the number N.

A prime number is a natural number that is only divisible by 1 and itself. 1 is not considered a prime number.

Prime or composite numbers? The last 10 numbers on which the prime factorization has been performed

Prime numbers. Composite numbers. The prime factorization of composite numbers (decomposing, breaking down numbers into prime factors)

  • The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
  • The fundamental theorem of arithmetic says that every integer larger than 1 can be written as a product of one or more prime numbers, in a way that is unique, except for the order of the prime factors.
  • The number 1 is not considered prime, so the first prime number is 2.
  • If the number 1 were considered a prime number, then the prime factorization of the number 15 could be written as: 15 = 3 × 5 OR 15 = 1 × 3 × 5 - these two representations would be considered different prime factorizations of the same number, so the theorem above would have no longer been valid.
  • The natural numbers that are evenly dividing only by 1 and themselves are called prime numbers.
  • Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and so on.
  • If a number is prime, it cannot be factored down to other prime factors, it is divisible only by 1 and itself - the number itself is called in this case an IMPROPER FACTOR (or an improper divisor). Some people also consider 1 as an improper factor.
  • A composite number is a natural number that has at least one factor (divisor) other than 1 and the number itself.
  • A composite number is also any natural number larger than 1 that is not a prime number.
  • Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26 and so on.
  • A prime number can't be factored down to other prime factors, but a number that is a composite can, as shown below:
  • Example 1: 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime, it's a composite number. 6 can be written as a product of factors in different ways, as: 6 = 1 × 6, or 6 = 1 × 2 × 3 or 6 = 2 × 3. But its prime factorization, regardless the order of the factors, is always: 6 = 2 × 3.
  • Example 2: 120 can be written as a product of factors in different ways, as 120 = 4 × 30 or 120 = 2 × 2 × 2 × 15 or 120 = 2 × 2 × 2 × 3 × 5. Its prime factorization is always: 120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5 - the last form of writing is the condensed form, with exponents, of the first form, the longer one.
  • * Note: 23 = 2 × 2 × 2 = 8. We are saying that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power.
  • Why is it important to know about the prime factorization of the numbers?
  • The prime factorization is useful when calculating the greatest common factor, GCF, of numbers (also called the greatest common divisor GCD, or the highest common factor, HCF).
  • GCF is needed when reducing (simplifying) fractions to the lowest terms.
  • The prime factorization comes in handy when calculating the least common multiple, LCM, of numbers - this is needed when adding or subtracting ordinary fractions, for example...
  • And the examples could continue (numbers divisibility, calculating all the factors of a number starting from its prime factorization, and so on...).
  • More examples of prime numbers:
  • 181 is divisible only by 181 and 1, so 181 is a prime number.
  • 2,341 is divisible only by 2,341 and 1, so 2,341 is a prime number.
  • 6,991 is divisible only by 6,991 and 1, so 6,991 is a prime number.
  • This is the list of all the prime numbers, from 1 up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
  • The prime numbers are used as basic blocks when building the prime factorization of the composite numbers. So we could say that the prime numbers really are the basic blocks of the composite numbers.