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The -digit numbers and are each multiples of . Which of the following could be the value of ?

Easy

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Recall the rule for divisibility by 3 . Then proceed to apply .

https://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_by_3_or_9 \( \\ \) We came to know that any number N (in base 10) is divisible by 3 iff the sum of all decimal digits of the number is divisible by 3 .Alternatively any number N (in base 10) is congruent to 0 mod 3 iff the sum of all decimal digits of the number is congruent to 0 mod 3 .

So , $$ 74A52B1 \ and \ \ 326AB4C \ \ are \ multiples \ of \ 3 \\ \Rightarrow 7+4+A+5+2+B+1 \equiv 0 (mod \ 3) \ and \ 3+2+6+A+B+4+C \equiv 0 (mod \ 3) \\ \Rightarrow A+B \equiv 2 (mod \ 3) \ and A+ B+C \equiv 0 (mod \ 3) \\ $$

$$ A+B \equiv 2 (mod \ 3) \ and \ \ A+ B+C \equiv 0 (mod 3) \\ \Rightarrow C \equiv 1 (mod \ 3) $$ So , C can be 1 , 4 or 7 . So option (A) is right .

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