Counting Three-Digit Numbers Divisible by 5

We need to find the total count of three-digit numbers that are exactly divisible by 5. Three-digit numbers range from 100 to 999.

Identifying the Range of Three-Digit Numbers Divisible by 5

A number is divisible by 5 if its last digit is either 0 or 5.

• The smallest three-digit number is 100. Is 100 divisible by 5? Yes, because it ends in 0. So, 100 is the first three-digit number divisible by 5.
• The largest three-digit number is 999. Is 999 divisible by 5? No, because it ends in 9. The largest three-digit number ending in 5 or 0 before 999 is 995. So, 995 is the last three-digit number divisible by 5.

The three-digit numbers divisible by 5 form an arithmetic progression (AP): 100, 105, 110, ..., 995.

In this arithmetic progression:

• The first term ($a_1$) is 100.
• The common difference ($d$) is 5 (since each subsequent term is 5 greater than the previous one).
• The last term ($a_n$) is 995.

Using Arithmetic Progression Formula to Find the Count

The formula for the n-th term of an arithmetic progression is:

\[a_n = a_1 + (n-1)d\]

Where:

• \(a_n\) is the last term
• \(a_1\) is the first term
• \(n\) is the number of terms (which is what we want to find)
• \(d\) is the common difference

We can plug in the values we identified:

\[995 = 100 + (n-1)5\]

Now, let's solve for \(n\):

Subtract 100 from both sides:

\[995 - 100 = (n-1)5\]

\[895 = (n-1)5\]

Divide both sides by 5:

\[\frac{895}{5} = n-1\]

\[179 = n-1\]

Add 1 to both sides:

\[179 + 1 = n\]

\[n = 180\]

So, there are 180 three-digit numbers that are divisible by 5.

Verification using an Alternative Method

Another way to think about this is to find the count of numbers divisible by 5 up to 999 and subtract the count of numbers divisible by 5 up to 99 (which are the two-digit and one-digit numbers).

• Count of numbers divisible by 5 up to 999: \(\lfloor \frac{999}{5} \rfloor = \lfloor 199.8 \rfloor = 199\)
• Count of numbers divisible by 5 up to 99: \(\lfloor \frac{99}{5} \rfloor = \lfloor 19.8 \rfloor = 19\)

The number of three-digit numbers divisible by 5 is the difference between these two counts:

\[\text{Count} = (\text{Numbers up to 999 divisible by 5}) - (\text{Numbers up to 99 divisible by 5})\]

\[\text{Count} = 199 - 19 = 180\]

Both methods yield the same result.

Thus, there are 180 three-digit numbers that are divisible by 5.