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For 10^n - 1 case: Check if the sum of digits taken n at a time from right is divisible by 999…9 (n digits). If yes then the original number is also divisible by 99…9 (n digits) Example - Is 6435 divisible by 99 ? 35 + 64 = 99. As per the above rule, 6435 is divisible by 99.

For 10^n + 1 case : Mark off the number in groups of n digits starting from the right, and add the n-digit groups together with alternating signs. If the sum is divisible by 10^n + 1 then the original number is also divisible by 10^n + 1. Eg: 4512276, (76 + 51) - (22 + 4) = 101, hence divisible by 101 Eg: 9533524, (524 + 9 ) - 533 = 0, hence divisible by 1001.

as a,b,c are coprimes to each other
GCD(a,b) = 1 = h
LCM(a,b)=a * b
F(a,b) = 1/(a * b)
now,
F [ 1 / F(a,b) , c ] = HCF [ 1/ab,c] / LCM[1/ab,c]
so, HCF [ 1/ab,c] = 1 / ab
LCM [ 1/ab , c] = c
so required function = 1/ab * 1/c = 1 / abc
and abc = 12....given
a = 1 , b = 3 , c = 4
so, a + b + c = 8

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