Whatisthegreatestpossibleperimeterof a right angledtriangle with integer side lengths if one of the sides has length 12?

You want the maximum of 12+a+bwhere a and bare such that 144+a^2=b^2. This is equivalent to (b−a)(b+a)=144. Making the list of all possible ways of expressing 144 as the product of two numbers both of which are even, you will find that the list is rather short. My answer comes from the fact that 144=2×72. For this decomposition, you get a=35 and b=37.