A right-angled triangle with integer side lengths has one side with length 605 . This side is neither the shortest side nor the longest side of the triangle. What is the maximum possible length of the shortest side of this triangle?
Since 605 is the middle side length of the right-angled triangle, we suppose that the side lengths of the triangle are $a, 605, c$ for integers $ac-a$. Note that $c>a$ and so $c+a>c-a>0$. We make a table of the possible values for $c+a$ and $c-a$, and use these to determine the possible values of $c$ and $a$

| $c+a$ | $c-a$ | $2 c=(c+a)+(c-a)$ | $c$ | $a=(c+a)-c$ |
| :---: | :---: | :---: | :---: | :---: |
| $5^{2} \cdot 11^{4}=366025$ | 1 | 366026 | 183013 | 103012 |
| $5 \cdot 11^{4}=73205$ | 5 | 73210 | 36605 | 36600 |
| $5^{2} \cdot 11^{3}=33275$ | 11 | 33286 | 16643 | 16632 |
| $11^{4}=14641$ | $5^{2}=25$ | 14666 | 7333 | 7308 |
| $5 \cdot 11^{3}=6655$ | $5 \cdot 11=55$ | 6710 | 3355 | 3300 |
| $5^{2} \cdot 11^{2}=3025$ | $11^{2}=121$ | 3146 | 1573 | 1452 |
| $11^{3}=1331$ | $5^{2} \cdot 11=275$ | 1606 | 803 | 528 |
| $5 \cdot 11^{2}=605$ | $5 \cdot 11^{2}=605$ | 1210 | 605 | 0 |

These are all of the possible factorizations of $605^{2}$, and so give all of the possible pairs $(a, c)$ that satisfy the equation.

Therefore, the maximum possible value of $a$ that is less than 605 is 528 .