1. Factorize 1040

Let's break 1040 down into its prime factors:
- $\(1040=104 \times 10\)$
- $\(104=2 \times 52=2 \times 2 \times 26=2 \times 2 \times 2 \times 13=2^3 \times 13\)$
- $\(10=2 \times 5\)$

Combining these:

$$
\(1040=2^4 \times 5^1 \times 13^1\)
$$

2. Analyze the Exponents

For $1040 k$ to be a perfect square, we look at the power of each prime:
- Base 2: The exponent is 4 (already even).
- Base 5: The exponent is 1 (needs to be at least 2 ).
- Base 13: The exponent is 1 (needs to be at least 2 ).

To make the exponents of 5 and 13 even, we must multiply by at least one more 5 and one more 13.
3. Calculate $k$

The smallest possible value for $k$ is:

$$
\(k=5 \times 13=65\)
$$


Let's check the result:

$$
\(1040 \times 65=\left(2^4 \times 5^1 \times 13^1\right) \times\left(5^1 \times 13^1\right)=2^4 \times 5^2 \times 13^2\)
$$


Since all exponents are even, this is $\(\left(2^2 \times 5 \times 13\right)^2=260^2\)$.