2001 AMC 12 Problem 6

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

Answer: E

Concepts:digits logical deduction casework

Difficulty rating: 1420

Small hint:

Four consecutive decreasing odd digits must be

9753

or

7531

Big hint:

The leftover odd digit sits in

ABC,

and with two even digits summing to

9

it must be

1

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.