self Maths Puzzle

plzzz explain remainder theorem

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self Other Question

A child was asked to add first few natural numbers(that is 1+2+3+4+5....)so long as his patience permitted . as he stopped he gave the sum as 575. when the teacher declred the result wrong , the child discovered he had missed one number in the sequence during addition . the no. he missed was? no of zeroes at the end of the expression (2!)^2!+4!^4!+8!^8!+9!^9!+10!^10!+11!^11!