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Math Help - Arithmetic Progression

  1. #1
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    Arithmetic 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?
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  2. #2
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    Quote Originally Posted by MichaelLight View Post
    Find the sum of the positive integers which are less than 150 and are not multiples of 5 or 7. Can anyone help me?
    This is the sum of the integers less than 150 minus the sum of multiples of 5 less than 150 minus the sum of multiples of 7 less than 150 plus multiples of 35 less than 150.

    Looks like a problem using the principle of inclusion/exclusion.

    CB
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  3. #3
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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 \lvert A\setminus (B\cup C)\rvert=\lvert A\rvert-\lvert B\cup C\rvert. Further, \lvert B\cup C\rvert=\lvert B\rvert+\lvert C\rvert-\lvert B\cap C\rvert. In all this, \setminus denotes set difference, \cup denotes union and \cap denotes intersection. Since 5 and 7 are coprime, 5\mid n and 7\mid n imply 35\mid n for all n. Here m\mid n means m divides n. Therefore, B\cap C = \{35, 70, \dots, 140\}.
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