Given a positive integer $n$, find the sum of all positive integers less than $10 n$ that are not multiples of 2 or 5.

# Solution.

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Let the sets be

$$
\begin{aligned}
A & =\{1,2, \ldots, 10 n\} \\
B & =\{2,4, \ldots, 2(5 n)\} \\
C & =\{5,10, \ldots, 5(2 n)\} \\
B \cap C & =\{10,20, \ldots, 10 n\}
\end{aligned}
$$

We are asked to find the sum of the elements of $A$ that are not in $B$ or $C$. The sums of the elements of each set are

$$
\begin{gathered}
\Sigma A=\frac{10 n(10 n+1)}{2}, \quad \Sigma B=2 \frac{5 n(5 n+1)}{2}, \quad \Sigma C=5 \frac{2 n(2 n+1)}{2}, \quad \text{and} \\
\Sigma(B \cap C)=\frac{10 n(10 n+1)}{2}
\end{gathered}
$$

The requested sum is

$$
\Sigma A-\Sigma B-\Sigma C+\Sigma(B \cap C)=20 n^{2}
$$

A2.-