Final answer:
There are 643 integers from 1 through 1,000 that are neither multiples of 4 nor multiples of 7, found by using the principle of inclusion-exclusion.
Step-by-step explanation:
To find out how many integers from 1 through 1,000 are neither multiples of 4 nor multiples of 7, we need to use the principle of inclusion-exclusion. First, we determine the number of multiples of each and then subtract the multiples of both 4 and 7 to avoid double-counting.
There are 250 multiples of 4 in 1,000 (since 1000 divided by 4 equals 250). Similarly, there are 142 multiples of 7 (since 1000 divided by 7 is approximately 142, but the exact number needs to be an integer, so we consider only up to 994, which is the largest multiple of 7 less than 1,000).
To find the numbers that are multiples of both 4 and 7, we find the Least Common Multiple (LCM) of 4 and 7, which is 28. There are 35 multiples of 28 within 1,000 (since 1000 divided by 28 is approximately 35).
We subtract the multiples of 4 and 7 from the total count of numbers 1 to 1,000 and then add back the multiples of 28 to correct for the over-counting:
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- Total count = 1,000
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- Multiples of 4 = 250
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- Multiples of 7 = 142
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- Multiples of both 4 and 7 (28) = 35
Using the principle of inclusion-exclusion, we calculate the count of numbers that are neither multiples of 4 nor multiples of 7:
Total count - (Multiples of 4 + Multiples of 7 - Multiples of 28) = 1,000 - (250 + 142 - 35) = 1,000 - 357 = 643.
Therefore, there are 643 integers from 1 through 1,000 that are neither multiples of 4 nor multiples of 7.