2001 AMC 12 Problem 6

Problem 6 of 25EasierNumber Theory

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.

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Solution

The four consecutive decreasing odd digits

GHIJ

are either

9753

7531,

leaving one odd digit (

1

9

) for

ABC.

Since

A + B + C = 9

and the other two digits of

ABC

are even, the odd digit must be

1

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,

420,

leaving the even pairs

\{2, 0\},

\{8, 0\},

\{8, 6\}

for

ABC.

Only

\{8, 0\}

sums to

8,

ABC = 810

and

A = 8.

Thus, the correct answer is E.