How many composite numbers are between 1 and 100?

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#

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.