2017 AMC 8 Problems/Problem 8
Contents
Problem
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its digits is 9.
This information allows Malcolm to determine Isabella's house number. What is its units digit?
Solution
Notice that (1) cannot be true. Otherwise, the number would have to be prime and be either even or divisible by 7. This only happens if the number is 2 or 7, neither of which are two-digit numbers, so we run into a contradiction. Thus, we must have (2), (3), and (4) be true. By (2), the 
-digit number is even, and thus, the digit in the tens place must be 
. The only even -digit number starting with and divisible by 
is 
, which has a units digit of 
Solution 3
Like solutions 1 and 2, Statement 1 can't be true because it would contradict both Statements 2 and 3. Therefore, the other three must be true. We know the following:
- If the number is even and has 9 as a digit, 9 must be the tens digit.
- Since it's divisible by both 2 and 7, the number is a multiple of 14.
The only multiple of 14 that has a tens digit of 9 is 98. Thus, our answer is
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See Also
| 2017 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
