Question

What are all the composite numbers between 31 and 51?

Answer 1

See a solution process below:

Explanation:

A composite number is: a whole number that can be divided evenly by numbers other than 1 or itself.

Because every even number is divisible by `2` all of the even numbers between 31 and 53 are part of the solution set for this problem:

`32, 34, 36, 38, 40, 42, 44, 46, 48. 50, 52`

Every number ending with `5` is evenly divisible by `5` so we can add these numbers to the solution set:

`32, 34, color(red)(35), 36, 38, 40, 42, 44, color(red)(45), 46, 48. 50, 52`

This leaves:

`33, 37, 39, 41, 43, 47, 49, 51`

The digits in `33`, `39` and `51` are evenly divisible by `3` so we can add them to the solution set:

`32, color(blue)(33), 34, color(red)(35), 36, 38, color(blue)(39), 40, 42, 44, color(red)(45), 46, 48. 50, color(blue)(51), 52`

This leaves:

`37, 41, 43, 47, 49`

`49` is `7^2` so this can be added to the solution set:

`32, color(blue)(33), 34, color(red)(35), 36, 38, color(blue)(39), 40, 42, 44, color(red)(45), 46, 48. color(orange)(49), 50, color(blue)(51), 52`

This leaves: `37, 41, 43, 47` These are all prime numbers and therefore not composite numbers so the solution is:

`32, color(blue)(33), 34, color(red)(35), 36, 38, color(blue)(39), 40, 42, 44, color(red)(45), 46, 48. color(orange)(49), 50, color(blue)(51), 52`

Answer 2

`32," "33," "34," "35," "36," "38," "39," "40," "42," "44," "45," "46," "48," "49," "50`

Explanation:

All natural numbers except `1` are either prime or composite,

There are fewer prime numbers than composite numbers.

So, in order to find all the composite numbers between `31 and 53`, it is easier to just remove all the prime numbers.

The prime numbers between `31 and 53` are:

`37," "41," "43," "47`

Removing these leaves us with:

`32," "33," "34," "35," "36," "cancel37," "38," "39," "40," "cancel41," "42," "cancel43," "44," "45," "46," "cancel47," "48," "49," "50`