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I stopped there because 16! would be too much to calculate. The 36th term would be (36^2)!, so the remainder would be 0 at this point based on the pattern, and the last term is 37!, so the remainder should be 1.
The way I solved it: (probably similar to the previous post).
1! + (2^2)! + 3! + (4^2)! + ... + 37!
= 1 + 4! + 6 + 16! + ... + 37!
= 25 + 6 + 16! + ... + 37!
Based on the above expression 25 would result in a remainder of 0, 6 would result in a remainder of 1 and every term after that all the way to 37! will always result in a remainder of 0, as 5 will be a part of each of them (due to the factorial). Hence remainder is 1.