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."
It is prime.
It is even.
It is divisible by .
One of its digits is .
This information allows Malcolm to determine Isabella's house number. What is its units digit?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are no two-digit even primes, so statements and cannot both be true. Also, no two-digit prime is divisible by , so statements and cannot both be true. Because there is only one false statement, it must be , so Isabella's house number is an even two-digit multiple of that has a digit of . The number is even, so the must be the tens digit. The only even multiple of between and is , so the units digit is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions