Not sure if this is the fastest way, but here is my approach.

Max number of the three can be 8.
Also 75 = odd number. So the three numbers that make up 75 can be:

1) even+even+odd
2) odd+odd+odd

1) Start with 2==>2^2 +4^2=4+16=20. Difference = 75-20=55 (in which case third number cannot be whole)
Next try 2^2 + 6^2 = 4 + 36=40
Next try 4^2+6^2=16+64=80 >75
Next try 8^2 = 64. Difference = 11. This cannot be sum of two square numbers.
So this combination of even+even+odd does not work.

2) The solution has to be all three odd numbers.
Start with 1==>1^2 + 3^2 + 5^2 = 35 < 75
Next try 1^2 + 5^2 + 7^2 = 75. (Other combinations would be 3^2 + 5 ^2 + 7^2 and this is >75)

For odd numbers we can try all combinations upto 7 and then stop because we already determined that the max number is 8.

Hence E.

Here's a quicker way of solving this question by Stuart,

https://www.beatthegmat.com/gmat-prep-t9319.html