2001 AMC 12 Problems/Problem 6

The following problem is from both the 2001 AMC 12 #6 and 2001 AMC 10 #13, so both problems redirect to this page.

Problem

A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find $A$.

$\text{(A)}\ 4\qquad \text{(B)}\ 5\qquad \text{(C)}\ 6\qquad \text{(D)}\ 7\qquad \text{(E)}\ 8$

Solution

The last four digits $\text{GHIJ}$ are either $9753$ or $7531$, and the other odd digit ($1$ or $9$) must be $A$, $B$, or $C$. Since $A + B + C = 9$, that digit must be $1$. Thus the sum of the two even digits in $\text{ABC}$ is $8$. $\text{DEF}$ must be $864$, $642$, or $420$, which respectively leave the pairs $2$ and $0$, $8$ and $0$, or $8$ and $6$, as the two even digits in $\text{ABC}$. Only $8$ and $0$ has sum $8$, so $\text{ABC}$ is $810$, and the required first digit is $\boxed{(\text{E})8}$.

See Also

2001 AMC 12 (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
2001 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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