2001 AMC 10 Problem 13

Problem 13 of 25IntermediateNumber 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 consecutive odd digits

GHIJ

are

9753

7531,

leaving one odd digit (

1

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,

420,

leaving even-digit 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.