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

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Notice that (1) cannot be true. Otherwise, the number would have to prime and either be 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) true. By (2), the <math>2</math>-digit number is even, and thus the digit in the tens place must be <math>9</math>. The only even <math>2</math>-digit number starting with <math>9</math> and divisible by <math>7</math> is <math>98</math>, which has a units digit of <math>\boxed{\textbf{(D)}\ 8}.</math>
 
Notice that (1) cannot be true. Otherwise, the number would have to prime and either be 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) true. By (2), the <math>2</math>-digit number is even, and thus the digit in the tens place must be <math>9</math>. The only even <math>2</math>-digit number starting with <math>9</math> and divisible by <math>7</math> is <math>98</math>, which has a units digit of <math>\boxed{\textbf{(D)}\ 8}.</math>
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==Solution==
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(1) Cannot be true, because only one of these four statements are true, and (1) states that the number is prime, which would make (2) and (3) false, which is not possible. Since the number is even, it must end with an even number. And since it is a two-digit number, the first digit must be 9 (according to statement 4). And out of <math>94</math>, <math>96</math>, and <math>98</math>, <math>98</math> is divisible by <math>7</math> (according to statement 3). So the answer is <math>\boxed{\textbf{(D)}\ 8}.</math>
  
 
==See Also==
 
==See Also==

Revision as of 22:04, 6 November 2020

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. Otherwise, the number would have to prime and either be 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) 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 $\boxed{\textbf{(D)}\ 8}.$

Solution

(1) Cannot be true, because only one of these four statements are true, and (1) states that the number is prime, which would make (2) and (3) false, which is not possible. Since the number is even, it must end with an even number. And since it is a two-digit number, the first digit must be 9 (according to statement 4). And out of $94$, $96$, and $98$, $98$ is divisible by $7$ (according to statement 3). So the answer is $\boxed{\textbf{(D)}\ 8}.$

See Also

2017 AMC 8 (ProblemsAnswer KeyResources)
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

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