Number Theory

Lesson

A prime number is a whole number greater than $1$1 that only has exactly $2$2 distinct factors, $1$1 and **itself**. For example, $3$3 is a prime number, since $3=1\times3=3\times1$3=1×3=3×1 and this is the only way we can write $3$3 as the product of two whole numbers. Similarly, $7=1\times7$7=1×7 and $31=1\times31$31=1×31, so $7$7 and $31$31 are also prime numbers.

A common misconception is that $1$1 is a prime number. Thinking back to the definition, that a prime number has exactly two **distinct** factors, we can see that $1$1 can only be written as a product of two numbers in the form $1\times1$1×1. Since these two numbers are the same (not distinct), then $1$1 cannot be a prime number.

A composite number is a number that has factors other than $1$1 and itself. That is, it can be divided evenly by a number some other whole number greater than $1$1.

Look at the picture below:

From the dots, we could make $1$1 group of $12$12, $2$2 groups or $6$6, $3$3 groups of $4$4, $4$4 groups of $3$3, $6$6 groups of $2$2 or even $12$12 groups of $1$1.

So the factors of $12$12 are $1,2,3,4,6$1,2,3,4,6 and $12$12.

Since there are so many ways that these $12$12 dots could be split up into equal groups, the number $12$12 is an example of a composite number.

Let's look at some others!

$6,14,20$6,14,20 and $21$21 are all composite numbers.

$6$6 can be written as $6\times1$6×1 or $2\times3$2×3.

$14$14 can be written as $1\times14$1×14 or $2\times7$2×7.

$20$20 can be written as $20\times1$20×1 or $10\times2$10×2 or $4\times5$4×5.

$21$21 can be written as $21\times1$21×1 or $3\times7$3×7.

Can you think of some other composite numbers?

This question is really asking, can the number be divided by another number other than $1$1 or itself? If it can, it is **composite**. If it can't, it's **prime**.

Is $17$17 prime or composite?

$17$17 can only be divided by $1$1 and $17$17, so it is prime.

Is $25$25 a prime number?

$25$25 can be divided by $1$1 and $25$25, but it can also be divided by $5$5 so the answer would be no, it is a composite number.

True or False?

$40$40 is a composite number.

True

AFalse

B

List the first prime number after $43$43 .

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True or False?

17 and 19 are twin primes.

True

AFalse

B