2017 AMC 8 Problem 8

Question

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?

4

6

7

8

9

Answer: D

Concepts:logical deduction divisibility

Difficulty rating: 1240

Small hint:

Since exactly three statements are true, only one statement can fail.

Big hint:

A two-digit number cannot be both prime and an even multiple greater than

2

, and it cannot be both prime and divisible by

7

.

Video solution:

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Written solution:

Statements

(1)

and

(2)

cannot both be true, because the only even prime is not two-digit. Statements

(1)

and

(3)

also cannot both be true, because a two-digit number divisible by

7

would not be prime. Since only one statement is false, statement

(1)

must be false, while statements

(2),

(3),

and

(4)

are true.

The house number is divisible by

2

and

7,

so it is divisible by

14.

Among the two-digit multiples of

14,

the only one with a digit of

9

is

98.

Therefore, the units digit is

8.

Thus, D is the correct answer.