How many integers from 1 through 1,000 are multiples of 4 or multiples of 7?
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How many integers from 1 through 1000 are neither multiples of 4 nor multiples of 7? j = 142 multiples of 7. 250 + 142 − 35 = 357 by the Inclusion-Exclusion Principle. Therefore the answer is 1000 − 357 = 643
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