the base-ten representation for 19! = 121,6T5,100,40M,832,000 then
H+M+T =12
19! has three 5s so it will end with three 0s.
Hence H = 0
19! = 121,6T5,100,40M,832,000 = 121,6T5,100,40M,832 * 1000
19! has many 2s. So M832 will be divisible by 16 (last 4 digits).
So M000 + 832 is divisible by 16. 832 is completely divisible by 16. Since 16*5 = 80, 8000 must be divisible by 16.
M = 8
19! = 121,6T5,100,408,832,000
Since 19! is divisible by 9, sum of all digits must be divisible by 9.
1+2+1+6+T+5+1+4+8+8+3+2 = T + 41
To make this divisible by 9, T must be 4.
H+M+T = 0 + 8 + 4 = 12
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