2001 AMC 10 Problem 13

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

The four consecutive odd digits

GHIJ

must be

9753

or

7531

Since

A+B+C=9

uses the leftover odd digit, that odd digit must be

1

The consecutive odd digits

GHIJ

are

9753

or

7531,

leaving one odd digit (

1

or

9

) for

A,

B,

C.

Since

A+B+C=9,

the odd digit there must be

1,

so the two even digits in

ABC

sum to

8.

The consecutive even digits

DEF

are

864,

642,

or

420,

leaving even-digit 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.

Problem 13 in other years