Study Guide: Prime Numbers
Vocabulary
| Term | Description |
|---|---|
| Prime Number | All whole numbers greater than 1 that have only two factors, the number itself and 1. For example, the only factors of 3 are 1 and 3, therefore 3 is prime. |
| Composite Number | A whole number greater than 1 that is the product of other whole numbers. |
| Factor | The numbers multiplied together to get another number. |
| Prime Factors | Factors of a product that are themselves prime. For example, the factors of 6 are 2 and 3. Both 2 and 3 are prime numbers. |
| Prime Factorization | The process ued to find which prime numbers are factors (multiply together) to get a number. |
| Factor Tree | A graphical way to factor numbers |
| Fundamental Theorem of Arithmetic | Every integer greater than 1 can be represented uniquely as a product of prime numbers. |
Example: What are the prime factors of 12?
Method 1
- Start the smallest prime factor, which in this case is 2.
$$ 12 \div 2 = 6 $$
- Check if the resulting number is composite. If it is, look for anther smallest prime factor. In this case, we can divide 6 by 2.
$$ 6 \div 2 = 3 $$
- Continue until only a prime factor remains. In this case 3 is a prime number, so we are finished.
$$ 12 = (2 \times 2 \times 3) = (2^2 \times 3) $$
Example 2: What is the prime factorization of 147?
- Try the prime number 2. Because 147 is not an even number, we know we can not divide 147 evenly by 2.
$$ 147 รท 2 = 73\frac{1}{2} $$
- The next smallest prime number to try is 3.
$$ 147 \div 3 = 49 $$
- That works. Now we need to factor 49. Three does not go evenly into 49. The next smallest prime number is 5. This also does not work, but 7 does.
$$ 49 \div 7 = 7 $$
- We’re done because all the factors that we found are prime numbers.
$$ 147 = (3 \times 7 \times 7) = 3 \times 7^2 $$
Method 2
Sometimes it’s easier to break a large number down into factors that you already know well, and then find the prime factors of each..
Example 3: What are the prime factors of 90?
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We can start with the composite numbers 9 and 10 because $ 90 = 9 \times 10 $.
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The prime factors of 9 are 3 and 3. ($9 = 3 \times 3$).
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The prime factors of 10 are 2 and 5. ($10 = 2 \times 5$)
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Therefore, the prime factors of 90 are 3, 3, 2 and 5. $(3 \times 3) \times (2 \times 5)$, or $(2 \times 3^2 \times 5)$
Why find Prime Factors?
Prime numbers are the basic building blocks of all numbers. This idea is useful when working with very large numbers, such as in Cryptography. Cryptography is the study of secret codes.
Prime Factorization is an important technique for those who make (or try to break) secret codes based on very large numbers. That is because factoring very large numbers is difficult, and can take even the fastest computers a long time to complete.
This is because every number has a unique set of prime factors. If you can find a number’s prime factors, you know everything about the number. This idea is so important it is called the Fundamental Theorem of Arithmetic. It is the reason why counting things actually works.
Questions
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What is the smallest prime number?
The smallest prime number is $2$. -
What is the largest prime number?
No one knows. -
How many prime numbers are there?
No one knows. It is possible that at some point all very larger numbers become composite, or not. -
What is the only even prime number?
The only even prime number is $2$, because all other even numbers have $2$ as a factor. -
Which two prime numbers are consecutive?
The only consecutive prime numbers are $2$ and $3$, because all other prime numbers are odd numbers. -
Is there a positive integer that is neither prime nor composite?
Yes, the number $1$.