# Math Help - Arithmetic Progression

1. ## 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?

2. Originally Posted by MichaelLight
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

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, $B\cap C = \{35, 70, \dots, 140\}$.