Composite number 10,471 prime factorization (decomposition), as a product of prime factors

Is 10,471 a composite or a prime number? What is its prime factorization? How to write it as a product of prime factors?

Integer prime factorization

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


* Positive integers that are only dividing by themselves and 1 are called prime numbers.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.



10,471 is not a prime, is a composite number.
10,471 can be written as a product of prime factors:
10,471 = 37 × 283

Note:

The prime numbers are the 'building blocks' of all numbers, except for 0 and 1.


All the composite numbers are 'made up' of prime numbers that are multiplied together.



How to factor a number, how to break it down to prime factors (prime factorization). Continued below.

Let's learn by having an example:
Take 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.

Final answer:
10,471 is not a prime, is a composite number.
Integer prime factorization, as a product of prime factors:
10,471 = 37 × 283

A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


Positive integers that are only dividing by themselves and 1 are called prime numbers.


More operations of this kind:


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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.

1 is not considered prime, so the first prime number is 2. If 1 were admitted as a prime, number 15 for example could be prime factorized as 3 × 5 and 1 × 3 × 5; these two representations would be considered different prime factorizations, so the theorem above would have to be modified.

Positive integers that are only dividing by themselves and by number 1 are called prime numbers. If a number is prime, it can not be factored down to other prime factors, it is divisible only by 1 and itself; the number itself is called an IMPROPER FACTOR (improper divisor). Some people also consider 1 as an improper factor.

A composite number is a positive integer that has at least one positive factor (divisor) other than 1 and the number itself. A composite number is also any positive integer larger than 1 that is not a prime number.

A prime number can't be factored down to prime factors, but a number that is a composite can be, as it is shown bellow:

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 factored in different ways, as 1 × 6, or 1 × 2 × 3, or 2 × 3; but its prime factorization is always: 6 = 2 × 3.

Example 2: 120 can be factored in different ways, as 4 × 30 or 2 × 2 × 2 × 15 or 2 × 2 × 2 × 3 × 5; its prime factorization is always: 120 = 23 × 3 × 5; this is the condensed form of writing, with exponents, of the longer: 120 = 2 × 2 × 2 × 3 × 5.

It is important to know about numbers prime factorization in order to calculate the greatest common factor GCF of numbers (also called the greatest common divizor GCD, or highest common factor, HCF) - GCF is needed when reducing (simplifying) fractions to the lowest terms, or to calculate the least common multiple, LCM - this is needed when adding or subtracting ordinary fractions...

Example of prime numbers: 2 is divisible only by 2 and 1, so 2 is a prime number; 13 is divisible only by 13 and 1, so 13 is a prime number;

Please have a look at all the prime numbers, from 2 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. Prime numbers are used as basic blocks when building the prime factorizations of the composite numbers.


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