The number of integers divisible by 7 depends entirely on the range of integers considered. There are infinitely many integers divisible by 7. However, within a specific range, the count is finite.
Examples and Calculations
Let's look at several examples to illustrate how to calculate the number of integers divisible by 7 within a given range:
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Integers between 1 and 100: There are 14 integers divisible by 7 (7, 14, 21,...,98). This is found by dividing 100 by 7 and taking the integer part of the result (100/7 ≈ 14.29, so 14).
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Integers between 60 and 600: As noted in the Quora link provided, this can be solved using an arithmetic progression. The first number divisible by 7 in this range is 63, and the last is 595. The formula to calculate the number of terms in an arithmetic sequence is: (last - first)/difference + 1. Therefore, the number of integers is (595 - 63)/7 + 1 = 76.
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Integers between 1 and 1000 (inclusive): Similar to the first example, we divide 1000 by 7 and take the integer part: 1000/7 ≈ 142.86, resulting in 142 integers divisible by 7. (This aligns with the example from zimmer.fresnostate.edu )
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Integers between 100 and 999 (inclusive): To find the number of integers divisible by 7 in this range, we can first find the number of integers divisible by 7 from 1 to 999 (142), then subtract the number of integers divisible by 7 from 1 to 99 (14). This gives us 142 - 14 = 128.
The key to solving these problems is to divide the upper bound of the range by 7 and round down to the nearest whole number. For a range with a lower bound greater than 0, adjust the calculation as shown in the last example.
