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

(Created page with "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 di...")
 
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Problem 8
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==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."
 
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."
  
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<math>\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
 
<math>\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
  
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<math>\textbf{Solution:}</math>
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==Solution==
  
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6, 8, and 4 are the only possibilities. The first digit is also 9. 98 is the only even number with the units digit of 9. <math>\boxed{D}</math> is our answer. It can not be prime because of the given conditions.
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Notice that (1) cannot be true, as we quickly see that we cannot have <math>2</math> of the <math>3</math> remaining conditions be true without running 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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==See Also==
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{{AMC8 box|year=2017|num-b=7|num-a=9}}
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{{MAA Notice}}

Revision as of 14:36, 22 November 2017

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 $\boxed{\textbf{(D)}\ 8}.$

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

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