2017 AMC 8 Problems/Problem 8: Difference between revisions
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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> | 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> | ||
~nukelauncher | |||
==See Also== | ==See Also== | ||
Revision as of 14:50, 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?
Solution
Notice that (1) cannot be true, as we quickly see that we cannot have 
of the 
remaining conditions be true without running into a contradiction. Thus, we must have (2), (3), and (4) true. By (2), the -digit number is even, and thus the digit in the tens place must be 
. The only even -digit number starting with and divisible by 
is 
, which has a units digit of 
~nukelauncher
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
