There are 112 three-digit numbers that are divisible by 8, forming an arithmetic progression.
Here's how we determine that:
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Identify the first three-digit number divisible by 8: The smallest three-digit number is 100. When 100 is divided by 8, the remainder is 4. Therefore, the first three-digit number divisible by 8 is 100 + (8 - 4) = 104.
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Identify the last three-digit number divisible by 8: The largest three-digit number is 999. When 999 is divided by 8, the remainder is 7. Therefore, the last three-digit number divisible by 8 is 999 - 7 = 992.
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Recognize the arithmetic progression: The three-digit numbers divisible by 8 form an arithmetic progression with:
- First term (a) = 104
- Common difference (d) = 8
- Last term (l) = 992
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Calculate the number of terms: We can use the formula for the nth term of an arithmetic progression: l = a + (n - 1)d. We need to find 'n' (the number of terms).
- 992 = 104 + (n - 1)8
- 992 - 104 = (n - 1)8
- 888 = (n - 1)8
- 888 / 8 = n - 1
- 111 = n - 1
- n = 111 + 1
- n = 112
Therefore, there are 112 three-digit numbers divisible by 8.
