The Factorial! Mega Challenge!
Consider a number which consists of digits such that the sum of the factorials of the digits is equal to the number .
Find the sum of the digits of .
The answer is 10.
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1 solution
145 = 1 + 24 + 120 = 1! + 4! + 5!
Explanation: no digit >6 can exist as 7! >1000
if 6 exists in representation, 6! = 720 ==> a digit >=7 must exist in hundreds digit, therefore 6 cannot exist
if 5 does not exist, the maximum number we can have is 3*4! = 72 < 100 which is not 3 digit, therefore 5 must appear at least once.
5 cannot be in the hundreds digit as the max we can have with no digits >= 6 is 3*5! = 360
From there, checking all digit combinations is probably fastest way of verifying that 145 is the only solution