2005 AMC 12B Problem 15

Question

The sum of four two-digit numbers is

221.

None of the eight digits is

0

and no two of them are the same. Which of the following is not included among the eight digits?

1

2

3

4

5

The eight distinct nonzero digits used have a total between

36

and

44

If the units digits sum to

U

and the tens digits to

T,

then

10T + U = 221,

so

U

ends in

1

The eight digits are distinct and chosen from

1

through

9,

whose total is

45.

So the eight used digits sum to between

45 - 9 = 36

and

45 - 1 = 44.

Let the four units digits sum to

U

and the four tens digits sum to

T.

Then

10T + U = 221,

so

U

ends in

1.

Since

1+2+3+4 = 10

\le U \le

6+7+8+9 = 30,

we have

U = 11

or

U = 21.

If

U = 11,

then

10T = 210,

so

T = 21

and the eight digits sum to

32,

which is below

36.

So

U = 21,

giving

T = 20

and total

41.

The missing digit is

45 - 41 = 4.

For example,

13 + 25 + 86 + 97 = 221.

Thus, the correct answer is D.

Problem 15 in other years