Difference between revisions of "2002 AMC 12B Problems/Problem 15"
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== Problem == | == Problem == | ||
− | How many four-digit numbers <math>N</math> have the property that the three-digit number obtained by removing the leftmost digit is one | + | How many four-digit numbers <math>N</math> have the property that the three-digit number obtained by removing the leftmost digit is one ninth of <math>N</math>? |
<math>\mathrm{(A)}\ 4 | <math>\mathrm{(A)}\ 4 | ||
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== Solution == | == Solution == | ||
Let <math>N = \overline{abcd} = 1000a + \overline{bcd}</math>, such that <math>\frac{N}{9} = \overline{bcd}</math>. Then <math>1000a + \overline{bcd} = 9\overline{bcd} \Longrightarrow 125a = \overline{bcd}</math>. Since <math>100 \le \overline{bcd} < 1000</math>, from <math>a = 1, \ldots, 7</math> we have <math>7</math> three-digit solutions, and the answer is <math>\mathrm{(D)}</math>. | Let <math>N = \overline{abcd} = 1000a + \overline{bcd}</math>, such that <math>\frac{N}{9} = \overline{bcd}</math>. Then <math>1000a + \overline{bcd} = 9\overline{bcd} \Longrightarrow 125a = \overline{bcd}</math>. Since <math>100 \le \overline{bcd} < 1000</math>, from <math>a = 1, \ldots, 7</math> we have <math>7</math> three-digit solutions, and the answer is <math>\mathrm{(D)}</math>. | ||
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+ | |||
+ | == Solution 2 == | ||
+ | Since N is a four digit number, assume WLOG that <math>N = 1000a + 100b + 10c + d</math>, where a is the thousands digit, b is the hundreds digit, c is the tens digit, and d is the ones digit. | ||
+ | Then, <math>\frac{1}{9}N = 100b + 10c + d</math>, so <math>N = 900b + 90c + 9d</math> | ||
+ | Set these equal to each other: | ||
+ | <cmath>1000a + 100b + 10c + d = 900b + 90c + 9d</cmath> | ||
+ | <cmath>1000a = 800b + 80c + 8d</cmath> | ||
+ | <cmath>1000a = 8(100b + 10c + d)</cmath> | ||
+ | Notice that <math>100b + 10c + d = N - 1000a</math>, thus: | ||
+ | <cmath>1000a = 8(N - 1000a)</cmath> | ||
+ | <cmath>1000a = 8N - 8000a</cmath> | ||
+ | <cmath>9000a = 8N</cmath> | ||
+ | <cmath>N = 1125a</cmath> | ||
+ | |||
+ | Go back to our first equation, in which we set <math>N = 1000a + 100b + 10c + d</math>, | ||
+ | Then: | ||
+ | <cmath>1125a = 1000a + 100b + 10c + d</cmath> | ||
+ | <cmath>125a = 100b + 10c + d</cmath> | ||
+ | The upper limit for the right hand side (RHS) is <math>999</math> (when <math>b = 9</math>, <math>c = 9</math>, and <math>d = 9</math>). | ||
+ | It's easy to prove that for an <math>a</math> there is only one combination of <math>b, c,</math> and <math>d</math> that can make the equation equal. Just think about the RHS as a three digit number <math>bcd</math>. There's one and only one way to create every three digit number with a certain combination of digits. | ||
+ | Thus, we test for how many as are in the domain set by the RHS. Since <math>125\cdot7 = 875</math> which is the largest <math>a</math> value, then <math>a</math> can be <math>1</math> through <math>7</math>, giving us the answer of <math>\boxed {D) 7}</math> | ||
+ | |||
+ | IronicNinja~ | ||
== See also == | == See also == | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 14:50, 3 April 2022
Contents
Problem
How many four-digit numbers have the property that the three-digit number obtained by removing the leftmost digit is one ninth of ?
Solution
Let , such that . Then . Since , from we have three-digit solutions, and the answer is .
Solution 2
Since N is a four digit number, assume WLOG that , where a is the thousands digit, b is the hundreds digit, c is the tens digit, and d is the ones digit. Then, , so Set these equal to each other: Notice that , thus:
Go back to our first equation, in which we set , Then: The upper limit for the right hand side (RHS) is (when , , and ). It's easy to prove that for an there is only one combination of and that can make the equation equal. Just think about the RHS as a three digit number . There's one and only one way to create every three digit number with a certain combination of digits. Thus, we test for how many as are in the domain set by the RHS. Since which is the largest value, then can be through , giving us the answer of
IronicNinja~
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 AMC 12 Problems and Solutions |
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