Question

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

$4$

$5$

$6$

$7$

$8$

Solution:

The four consecutive decreasing odd digits $GHIJ$ are either $9753$ or $7531,$ leaving one odd digit ( $1$ or $9$ ) for $ABC.$

Since $A + B + C = 9$ and the other two digits of $ABC$ are even, the odd digit must be $1$ (a $9$ would force the two even digits to sum to $0$ ). So the two even digits sum to $8.$

The three consecutive decreasing even digits $DEF$ are $864,$ $642,$ or $420,$ leaving the even pairs $\{2, 0\},$ $\{8, 0\},$ or $\{8, 6\}$ for $ABC.$ Only $\{8, 0\}$ sums to $8,$ so $ABC = 810$ and $A = 8.$

Thus, the correct answer is E.

2001 AMC 12 Problem 6

Solution:

Problem 6 in Other Years