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?

$1$

$2$

$3$

$4$

$5$

Solution:

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.

2005 AMC 12B Problem 15

Solution:

Problem 15 in Other Years