Question

How many composite numbers are between 1 and 100?

Answer 1

`74`

Explanation:

Note that the following answer assumes that "between 1 and 100" is intended to include `1` and `100`, following common English usage.

Start with the numbers `1,2,3,...100`

• `1` is not composite, because it's a unit, so that leaves `99` other numbers.

• The numbers `4=2^2, 6, 8, ... , 100` are all divisible by `2`, so composite. There are `(100-4)/2+1 = 49` of these, leaving `99-49 = 50` other numbers.

• The numbers `9=3^2, 15, 21,..., 99` are all divisible by `3` and not by `2`. There are `(99-9)/6+1 = 16` of these, leaving `50-16 = 34` other numbers.

• The numbers `25=5^2, 35, 55, 65, 85, 95` are all divisible by `5` and not by `2` or `3`. There are `6` of these, leaving `34-6 = 28` other numbers.

• The numbers `49=7^2, 77, 91` are divisible by `7` and not by `2`, `3` or `5`. There are `3` of these, leaving `28-3 = 25` other numbers.

These `25` numbers must be prime, since `11^2 = 121 > 100`

So the total number of composite numbers is:

`49+16+6+3 = 74`

Answer 2

There are `73` composite numbers less than `100`

Explanation:

There are `100` numbers from `1 " to " 100`

However, the question specifies numbers BETWEEN `1 and 100`, so these two numbers are eliminated from the total.

So we are left with `98` numbers to work with.

All the numbers between `1 and 100` are either prime or composite.
(The only number that is neither is `1` which has already been excluded)

There are `25` prime numbers less than `100`.

This is pretty easy to check by just counting them, but it is a small fact that is worth knowing.

Therefore, if `25` of the `98` numbers are prime, it means that all the rest are composite:

`98-25 =73` composite numbers.