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2014 AMC 8 Problems/Problem 21

Revision as of 12:47, 5 September 2015 by Zacker205 (talk | contribs) (→‎See Also)

Problem

The $7$-digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

Solution

The sum of a number's digits $\mod{3}$ is congruent to the number $\pmod{3}$. $74A52B1 \mod{3}$ must be congruent to 0, since it is divisible by 3. Therefore, $7+4+A+5+2+B+1 \mod{3}$ is also congruent to 0. $7+4+5+2+1 \equiv 1 \pmod{3}$, so $A+B\equiv 2 \pmod{3}$. As we know, $326AB4C\equiv 0 \pmod{3}$, so $3+2+6+A+B+4+C =15+A+B+C\equiv 0 \pmod{3}$, and therefore $A+B+C\equiv 0 \pmod{3}$. We can substitute 2 for $A+B$, so $2+C\equiv 0 \pmod{3}$, and therefore $C\equiv 1\pmod{3}$. This means that C can be 1, 4, or 7, but the only one of those that is an answer choice is $\textbf{(A) }1$.

See Also

[[{{{year}}} AMC 8]] ([[{{{year}}} AMC 8 Problems|Problems]] • [[{{{year}}} AMC 8 Answer Key|Answer Key]] • Resources)
Preceded by
[[{{{year}}} AMC 8 Problems/Problem 20|Problem 20]] 		Followed by
[[{{{year}}} AMC 8 Problems/Problem 22|Problem 22]]
[[{{{year}}} AMC 8 Problems/Problem 1|1]] • [[{{{year}}} AMC 8 Problems/Problem 2|2]] • [[{{{year}}} AMC 8 Problems/Problem 3|3]] • [[{{{year}}} AMC 8 Problems/Problem 4|4]] • [[{{{year}}} AMC 8 Problems/Problem 5|5]] • [[{{{year}}} AMC 8 Problems/Problem 6|6]] • [[{{{year}}} AMC 8 Problems/Problem 7|7]] • [[{{{year}}} AMC 8 Problems/Problem 8|8]] • [[{{{year}}} AMC 8 Problems/Problem 9|9]] • [[{{{year}}} AMC 8 Problems/Problem 10|10]] • [[{{{year}}} AMC 8 Problems/Problem 11|11]] • [[{{{year}}} AMC 8 Problems/Problem 12|12]] • [[{{{year}}} AMC 8 Problems/Problem 13|13]] • [[{{{year}}} AMC 8 Problems/Problem 14|14]] • [[{{{year}}} AMC 8 Problems/Problem 15|15]] • [[{{{year}}} AMC 8 Problems/Problem 16|16]] • [[{{{year}}} AMC 8 Problems/Problem 17|17]] • [[{{{year}}} AMC 8 Problems/Problem 18|18]] • [[{{{year}}} AMC 8 Problems/Problem 19|19]] • [[{{{year}}} AMC 8 Problems/Problem 20|20]] • [[{{{year}}} AMC 8 Problems/Problem 21|21]] • [[{{{year}}} AMC 8 Problems/Problem 22|22]] • [[{{{year}}} AMC 8 Problems/Problem 23|23]] • [[{{{year}}} AMC 8 Problems/Problem 24|24]] • [[{{{year}}} AMC 8 Problems/Problem 25|25]]
All AJHSME/AMC 8 Problems and Solutions

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