<s>[INST] Find the remainder of $\frac{1! +2!+\, \dots\, +95! }{15}$. I think I found my answer but I am looking for better ones
 [/INST] LHN = 1 + 2 + (2.3) + ................................ + (2.3.4.....95)
    = 1 + 2 + (2.3) + (2.3.4) + (2.3.4.5) + .........+ (2.3.4.....95)
    = 1 + 2 + (6) + (24) + (15.8) +...............+ (2.3.4.....95)
    = ( 1 + 2 +   6    +   24  ) + (15.8) + (15.48)+............+ (2.3.4.....95)  
So,
   LHN modulo 15
 =  { (33) + (15.8) + (15.8.6)+............+ (2.3.4.....95) } modulo 15
=  (33 modulo 15 ) + {  15(8 + 8.6 + 8.6.7 + 8.6.7.8 + .........+ (8.6.7.8....95) } modulo  15
 = (33 % 15 ) + { 15 x SOMETHING } % 15
 = 3 + 0 = 3
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