# Let #5a+12b# and #12a+5b# be the side lengths of a right-angled triangle and #13a+kb# be the hypotenuse, where #a#, #b# and #k# are positive integers. How do you find the smallest possible value of #k# and the smallest values of #a# and #b# for that #k#?

##### 1 Answer

#### Answer:

#### Explanation:

By Pythagoras' theorem, we have:

#(13a+kb)^2 = (5a+12b)^2+(12a+5b)^2#

That is:

#169a^2+26kab+k^2b^2 = 25a^2+120ab+144b^2+144a^2+120ab+25b^2#

#color(white)(169a^2+26kab+k^2b^2) = 169a^2+240ab+169b^2#

Subtract the left hand side from both ends to find:

#0 = (240-26k)ab + (169-k^2)b^2#

#color(white)(0) = b((240-26k)a+(169-k^2)b)#

Since

#(240-26k)a+(169-k^2)b = 0#

Then since

When

When

So the minimum possible value of

Then:

#-20a+69b = 0#

Then since