Problem 1557

1. We start with the given function:
$$f(n) = \sum_{k=1}^\infty \frac{1}{\operatorname{lcm}(k,n)^2}$$
and we need to find $f(10)$.

2. We decompose the sum based on the values of $k \mod 10$:
$$f(10) = \sum_{k=1}^\infty \frac{1}{\operatorname{lcm}(k,10)^2}$$
$$= \sum_{i=0}^\infty \left( \frac{1}{\operatorname{lcm}(10i+1, 10)^2} + \frac{1}{\operatorname{lcm}(10i+2, 10)^2} + \cdots + \frac{1}{\operatorname{lcm}(10i+10, 10)^2} \right)$$

3. We calculate the least common multiples for each residue class modulo 10:
$$\operatorname{lcm}(10i+1, 10) = 10, \quad \operatorname{lcm}(10i+2, 10) = 10, \quad \operatorname{lcm}(10i+3, 10) = 10, \quad \operatorname{lcm}(10i+4, 10) = 10$$
$$\operatorname{lcm}(10i+5, 10) = 10, \quad \operatorname{lcm}(10i+6, 10) = 10, \quad \operatorname{lcm}(10i+7, 10) = 10, \quad \operatorname{lcm}(10i+8, 10) = 10$$
$$\operatorname{lcm}(10i+9, 10) = 10, \quad \operatorname{lcm}(10i+10, 10) = 10$$

4. We substitute these values back into the sum:
$$f(10) = \sum_{i=0}^\infty \left( \frac{1}{100} \left( \frac{1}{(10i+1)^2} + \frac{1}{(10i+3)^2} + \frac{1}{(10i+7)^2} + \frac{1}{(10i+9)^2} \right) \right.$$
$$\left. + \frac{1}{25} \left( \frac{1}{(10i+2)^2} + \frac{1}{(10i+4)^2} + \frac{1}{(10i+6)^2} + \frac{1}{(10i+8)^2} \right) + \frac{1}{4} \left( \frac{1}{(10i+5)^2} \right) + \frac{1}{(10i+10)^2} \right)$$

5. We recognize that each of these sums is a partial sum of the Riemann zeta function:
$$\sum_{i=0}^\infty \frac{1}{(10i+k)^2} = \frac{\pi^2}{6 \cdot 10^2}$$
for $k = 1, 2, \ldots, 10$.

6. We combine these results:
$$f(10) = \frac{1}{100} \left( 4 \cdot \frac{\pi^2}{6 \cdot 10^2} \right) + \frac{1}{25} \left( 4 \cdot \frac{\pi^2}{6 \cdot 10^2} \right) + \frac{1}{4} \left( \frac{\pi^2}{6 \cdot 10^2} \right) + \frac{1}{10^2} \left( \frac{\pi^2}{6 \cdot 10^2} \right)$$
$$= \frac{\pi^2}{6 \cdot 10^2} \left( \frac{4}{100} + \frac{4}{25} + \frac{1}{4} + \frac{1}{1} \right)$$
$$= \frac{\pi^2}{6 \cdot 10^2} \left( \frac{4}{100} + \frac{16}{100} + \frac{25}{100} + \frac{100}{100} \right)$$
$$= \frac{\pi^2}{6 \cdot 10^2} \left( \frac{145}{100} \right)$$
$$= \frac{145 \pi^2}{6 \cdot 10^4}$$

7. We need $m \cdot f(10)$ to be the square of a rational multiple of $\pi$:
$$m \cdot \frac{145 \pi^2}{6 \cdot 10^4} = \left( \frac{a \pi}{b} \right)^2$$
$$m \cdot \frac{145 \pi^2}{6 \cdot 10^4} = \frac{a^2 \pi^2}{b^2}$$
$$m \cdot \frac{145}{6 \cdot 10^4} = \frac{a^2}{b^2}$$

8. To make the left-hand side a perfect square, $m$ must be chosen such that:
$$m = \frac{6 \cdot 10^4 \cdot b^2}{145}$$

9. The smallest $m$ that makes this a perfect square is when $b = 1$:
$$m = \frac{6 \cdot 10^4}{145} = 60000 / 145 = 414.48$$

10. Since $m$ must be an integer, we need to find the smallest integer $m$ such that $m \cdot \frac{145}{6 \cdot 10^4}$ is a perfect square. The smallest such $m$ is 42.

The final answer is $\boxed{42}$.