How do you list all the prime numbers between 80 and 90?

In division testing you may skip `2` now, we can also skip `5` if the number doesn't end in `5` or `0`

`81//3=27` comes out evenly

`83//3=27` and 2 left
`83//7=11` and 6 left
We don't have to test any further: 83 is prime

`85//5=19` comes out evenly

`87//3=29` comes out evenly

`89//3=29` and 2 left
`89//7=12` and 5 left
We don't have to test any further: 89 is prime

Note : We didn't have to test for the next prime 11, because if the number has a factor of 11 or higher, it also has a factor smaller than 11, and we would have found that. The highest prime you test for is smaller/equal to the square root of the number.

To list the prime numbers between `80 and 90` we can use the rules of divisibility as a start;

`80, 81," "82," "83," "84," "85," "86," "87," "88," "89," "90`

All even numbers are divisible 2, they are not prime. apart from `2`
This leaves all the odd numbers.

`81," "83," "85," "87," "89,`

A number ending in `5` is divisible by `5`

`81," "83," "87," "89,`

If the sum of the digits is divisible by `3`, `3` is a factor

`81" "rarr 8+1=9." "` 81 is not prime

`83" "rarr 8+3 = 11" "` 83 might be prime

`87" "rarr 8+7=15" "` 87 is not prime

`89" "rarr 8+9 = 17" "`89 might be prime

As factors are always in pairs we only need to consider factors less than the square roots.

`sqrt83 = 9.....`

`2,3,4,5,6,7,8,9` are not factors: `83` is prime

`sqrt89 = 9......`

`2,3,4,5,6,7,8,9` are not factors: `89` is prime