# Prime Factorization

## Definition

The Fundamental Theorem of Arithmetic states that every positive integer can factored into primes uniquely, and in only one way (ignoring the order of multiplication). That is to say, for any positive whole number, $n$, there exists only one possible prime factorization.

For example:

$588 = 2^2 \times 3 \times 7^2$

and it is not possible to write 588 as some other product of prime factors (e.g., $588 \neq 2^2 \times 11 \times 13$ ).

## Technique

### What is the sum of the prime factors of 1190?

Since $1190 = 10 \times 119 = 2 \times 5 \times 119$, the problem reduces to factoring 119. Factoring large integers is a very hard problem, but 119 is small enough that we can find the answer by inspection. $119 = 7 \times 17$, so $1190 = 2 \times 5 \times 7 \times 17$.

$2 + 5 + 7 + 17 = 31$, which is our answer.

## Application and Extensions

### What is the least common multiple of 14, 15, and 20?

It will be helpful to look at these numbers in their factored form.

\begin{aligned} 14 &= 2 \times 7 \\ 15 &= 3 \times 5 \\ 20 &= 2^2 \times 5 \end{aligned}

Let $M = 14a = 15b = 20c$ be the smallest number that satisfies this equality for positive integer values of $a$, $b$, and $c$. Since $\frac{M}{14}=a$, and $a$ is an integer, $M$ must contain both 2 and 7 as factors. Reasoning similarly for 15, and 20 makes clear that $M = 2^2 \times 3 \times 5 \times 7 = 420 = (30)(14) = (28)(15) = (21)(20)$.

Finally, $M = 420$ must be the smallest such number because if we remove any of the stated factors, it will no longer be an integer multiple of one of: 14, 15, or 20.

Note by Arron Kau
5 years, 10 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

i want an testing exercise

- 5 years, 9 months ago

You might try the Practice section of the site. There's a "prime factorization" skill early in the map.

Staff - 5 years, 9 months ago