There are exactly 10 three-digit numbers that are divisible by 87.
Finding Three-Digit Numbers Divisible by 87
To determine how many three-digit numbers are divisible by 87, we need to find the smallest and largest three-digit multiples of 87.
- Smallest Three-Digit Multiple of 87: The first three-digit number divisible by 87 is 174.
- Largest Three-Digit Multiple of 87: The largest three-digit number is 999. Dividing 999 by 87, we get approximately 11.48. This means that 11 87 is the largest three-digit multiple of 87. Calculating 11 87, we get 957.
Arithmetic Progression Approach
The multiples of 87 form an arithmetic progression.
- The first term (a) is 174.
- The common difference (d) is 87.
- The last term (l) is 957.
We can use the formula for the nth term of an arithmetic progression:
l = a + (n - 1)d
Where:
- l = last term (957)
- a = first term (174)
- n = number of terms (what we want to find)
- d = common difference (87)
Plugging in the values:
957 = 174 + (n - 1)87
Solving for n:
957 - 174 = (n - 1)87
783 = (n - 1)87
783 / 87 = n - 1
9 = n - 1
n = 10
Therefore, there are 10 three-digit numbers divisible by 87.
