Number Theory

Lesson

When we're thinking about factors we need to think about dividing and multiplying numbers. Every number will have at least two factors, $1$1 and the number itself.

If we're finding all the factors of a number we are trying to find all the numbers we can multiply together to make that specific number.

So, if we want to find all the factors of $12$12 we need to find all the numbers we can multiply together to get $12$12. Let's find the factors of $12$12

$1$1 | $\times$× | $12$12 | $=$= | $12$12 |

$2$2 | $\times$× | $6$6 | $=$= | $12$12 |

$3$3 | $\times$× | $4$4 | $=$= | $12$12 |

Remember that $1$1 $\times$× $12$12 is the same as $12$12 $\times$× $1$1.

In this problem the numbers we are multiplying together are the factors of $12$12. So we can list the factors of $12$12, they are:

$1$1, $2$2, $3$3, $4$4, $6$6 and $12$12.

Let's try a harder one.

What possible values (up to $100$100) would have factors of $3$3 and $11$11?

In a question like this we might start with all the multiples of $3$3.

So counting them it would go

$3$3, $6$6, $9$9, $12$12, $15$15... all the way to $99$99!

A better way to find the answer to this question would be to use the bigger number. That way we don't have to count check as many numbers to see if they're divisible by $3$3 as well. When we have multiple numbers it's always easiest to think about the biggest number first.

So, starting from $11$11, let's see which multiples of $11$11 are also divisible by $3$3.

Multiples of $11$11 | Is it divisible by $3$3? | Numbers that have both $3$3 and $11$11 as factors |
---|---|---|

$11$11 | no | |

$22$22 | no | |

$33$33 | yes | $33$33 |

$44$44 | no | |

$55$55 | no | |

$66$66 | yes | $66$66 |

$77$77 | no | |

$88$88 | no | |

$99$99 | yes | $99$99 |

So now we have found three numbers that have factors of both $11$11 and $3$3.

They are $33$33, $66$66 and $99$99.

What are the factors of $42$42? Separate the factors with commas.

What factor of $30$30, other than $1$1, is also a factor of $27$27 and $21$21?

What numbers from $1$1 to $100$100 have factors of $3$3, $5$5 and $10$10?

Write each number on the same line, separated by commas.