# How many composite numbers are between 1 and 100?

May 27, 2017

$74$

#### 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 $1 , 2 , 3 , \ldots 100$

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

• The numbers $4 = {2}^{2} , 6 , 8 , \ldots , 100$ are all divisible by $2$, so composite. There are $\frac{100 - 4}{2} + 1 = 49$ of these, leaving $99 - 49 = 50$ other numbers.

• The numbers $9 = {3}^{2} , 15 , 21 , \ldots , 99$ are all divisible by $3$ and not by $2$. There are $\frac{99 - 9}{6} + 1 = 16$ 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:

$49 + 16 + 6 + 3 = 74$

May 27, 2017

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

#### Explanation:

There are $100$ numbers from $1 \text{ to } 100$

However, the question specifies numbers BETWEEN $1 \mathmr{and} 100$, so these two numbers are eliminated from the total.

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

All the numbers between $1 \mathmr{and} 100$ 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.