Find the sum of the positive integers which are less than 150 and are not multiples of 5 or 7. Can anyone help me?

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- Feb 28th 2011, 01:29 AMMichaelLightArithmetic Progression
Find the sum of the positive integers which are less than 150 and are not multiples of 5 or 7. Can anyone help me?

- Feb 28th 2011, 02:04 AMCaptainBlack
- Feb 28th 2011, 02:11 AMemakarov
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You can use Inclusion-Exclusion Principle. (You don't know much about it; it's mostly common sense.) Let A = {1, ..., 149}, B = {5, 10, 15, ..., 145} (multiples of 5) and C = {7, 14, 21, ..., 147} (multiples of 7). Let |X| denote the number of elements in the set X. Then the answer is . Further, . In all this, denotes set difference, denotes union and denotes intersection. Since 5 and 7 are coprime, and imply for all . Here means divides . Therefore, .