The sum of the areas of two squares is 85. If the sides of both squares have integer lengths, what is the least possible value for the length of a side of the smaller square?

Answer

To solve the problem, we need to find the least possible value for the length of a side of the smaller square, given that the sum of the areas of two squares is 85 and both squares have integer side lengths.

### Step-by-Step Solution:

1. **Define Variables**: Let the side lengths of the two squares be \( A \) (the smaller square) and \( B \) (the larger square).

2. **Set Up the Equation**: The areas of the squares can be expressed as:
\[ A^2 + B^2 = 85 \]
where \( A^2 \) is the area of the smaller square and \( B^2 \) is the area of the larger square.

3. **Find Integer Values for A**: Since \( A \) and \( B \) must be integers, we will test integer values for \( A \) starting from the smallest integer (1) and increasing.

4. **Check \( A = 1 \)**:
\[ 1^2 + B^2 = 85 \implies B^2 = 85 - 1 = 84 \]
\( B = \sqrt{84} \) which is not an integer.

5. **Check \( A = 2 \)**:
\[ 2^2 + B^2 = 85 \implies B^2 = 85 - 4 = 81 \]
\( B = \sqrt{81} = 9 \), which is an integer.

6. **Check \( A = 3 \)**:
\[ 3^2 + B^2 = 85 \implies B^2 = 85 - 9 = 76 \]
\( B = \sqrt{76} \) which is not an integer.

7. **Check \( A = 4 \)**:
\[ 4^2 + B^2 = 85 \implies B^2 = 85 - 16 = 69 \]
\( B = \sqrt{69} \) which is not an integer.

8. **Check \( A = 5 \)**:
\[ 5^2 + B^2 = 85 \implies B^2 = 85 - 25 = 60 \]
\( B = \sqrt{60} \) which is not an integer.

9. **Check \( A = 6 \)**:
\[ 6^2 + B^2 = 85 \implies B^2 = 85 - 36 = 49 \]
\( B = \sqrt{49} = 7 \), which is an integer.

10. **Check \( A = 7 \)**:
\[ 7^2 + B^2 = 85 \implies B^2 = 85 - 49 = 36 \]
\( B = \sqrt{36} = 6 \), which is an integer.

11. **Check \( A = 8 \)**:
\[ 8^2 + B^2 = 85 \implies B^2 = 85 - 64 = 21 \]
\( B = \sqrt{21} \) which is not an integer.

12. **Check \( A = 9 \)**:
\[ 9^2 + B^2 = 85 \implies B^2 = 85 - 81 = 4 \]
\( B = \sqrt{4} = 2 \), which is an integer.

13. **Conclusion**: The least possible value for the length of a side of the smaller square \( A \) is 2, as it satisfies the condition of both squares having integer lengths and the sum of their areas being 85.

### Final Answer:
The least possible value for the length of a side of the smaller square is \( \boxed{2} \).