# Difference between revisions of "2017 AMC 8 Problems/Problem 8"

## Problem 8

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? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

## Solution

Notice that (1) cannot be true, as we quickly see that we cannot have $2$ of the $3$ remaining conditions be true without running into a contradiction. Thus, we must have (2), (3), and (4) true. By (2), the $2$-digit number is even, and thus the digit in the tens place must be $9$. The only even $2$-digit number starting with $9$ and divisible by $7$ is $98$, which has a units digit of $\boxed{\textbf{(D)}\ 8}.$

~nukelauncher

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 