How many three-digit numbers are divisible by 5 or 9?

There are 260 three-digit numbers that are divisible by 5 or 9.

Here's how we can calculate that:

1. Three-Digit Numbers Divisible by 5:

• The first three-digit number divisible by 5 is 100.
• The last three-digit number divisible by 5 is 995.
• To find the total number of three-digit numbers divisible by 5, we can use the arithmetic sequence formula: a_n = a₁ + (n-1)d, where a_n is the last term, a₁ is the first term, n is the number of terms, and d is the common difference.
• In this case, 995 = 100 + (n - 1)5.
• Solving for n: 895 = (n-1)5 => 179 = n-1 => n = 180.
• So, there are 180 three-digit numbers divisible by 5.

2. Three-Digit Numbers Divisible by 9:

• The first three-digit number divisible by 9 is 108.
• The last three-digit number divisible by 9 is 999.
• Using the arithmetic sequence formula: 999 = 108 + (n - 1)9.
• Solving for n: 891 = (n-1)9 => 99 = n-1 => n = 100.
• So, there are 100 three-digit numbers divisible by 9.

3. Three-Digit Numbers Divisible by Both 5 and 9 (i.e., divisible by 45):

• We need to find numbers divisible by the least common multiple (LCM) of 5 and 9, which is 45.
• The first three-digit number divisible by 45 is 135.
• The last three-digit number divisible by 45 is 990.
• Using the arithmetic sequence formula: 990 = 135 + (n - 1)45.
• Solving for n: 855 = (n-1)45 => 19 = n-1 => n = 20.
• So, there are 20 three-digit numbers divisible by 45.

4. Using the Inclusion-Exclusion Principle:

To find the number of three-digit numbers divisible by 5 or 9, we add the number divisible by 5 and the number divisible by 9, and then subtract the number divisible by both (to avoid double-counting).

• Total = (Divisible by 5) + (Divisible by 9) - (Divisible by both 5 and 9)
• Total = 180 + 100 - 20 = 260

Therefore, there are 260 three-digit numbers that are divisible by 5 or 9.