Difference between revisions of "2002 AMC 12B Problems/Problem 4"
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\qquad\mathrm{(E)}\ n > 84</math> | \qquad\mathrm{(E)}\ n > 84</math> | ||
− | == Solution == | + | == Solution 1== |
− | Since <math>\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}</math>, | + | Since <math>\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}</math>, <math>0 < \lim_{n \rightarrow \infty} \left(\frac{41}{42} + \frac{1}{n}\right) < \frac {41}{42} + \frac 1n < \frac{41}{42} + \frac 11 < 2</math> |
− | < | + | From which it follows that <math>\frac{41}{42} + \frac 1n = 1</math> and <math>n = 42</math>. The only answer choice that is not true is <math>\boxed{\mathrm{(E)}\ n>84}</math>. |
− | + | == Solution 2 (no limits)== | |
+ | Since <math>\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}</math>, it is very clear that <math>n=42</math> makes the expression an integer. Because <math>n</math> is a positive integer, <math>\frac{1}{n}</math> must be less than or equal to <math>1</math>. Thus, the only integer the expression can take is <math>1</math>, which means the only value for <math>n</math> is <math>42</math>. Thus <math>\boxed{\mathrm{(E)}\ n>84}</math> | ||
+ | |||
+ | ~superagh | ||
+ | |||
+ | == Solution 3(similiar to solution2)== | ||
+ | Cross multiplying and adding the fraction we get the fraction to be equal to <math>\frac {41n + 42}{42n}</math>, This value has to be an integer. This implies, | ||
+ | |||
+ | <math>42n|(41n+42)</math>. | ||
+ | |||
+ | => <math>42|(41n + 42)</math> | ||
+ | but <math>42|42</math>, hence <math>42|41n</math> | ||
+ | but <math>42</math> does not divides <math>41</math>, | ||
+ | <math>42|n</math> -(1) | ||
+ | |||
+ | => <math>n|(41n+42)</math> | ||
+ | but <math>n|41n</math>, | ||
+ | <math>n|42</math> -(2) | ||
+ | |||
+ | from (1) and (2) we get that n=42. | ||
+ | Comparing this with the options, we see that option E is the incorrect statement and hence E is the answer. | ||
+ | |||
+ | ~rudolf1279 | ||
== See also == | == See also == | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 10:54, 9 August 2022
- The following problem is from both the 2002 AMC 12B #4 and 2002 AMC 10B #7, so both problems redirect to this page.
Contents
Problem
Let be a positive integer such that
is an integer. Which of the following statements is not true:
Solution 1
Since ,
From which it follows that and
. The only answer choice that is not true is
.
Solution 2 (no limits)
Since , it is very clear that
makes the expression an integer. Because
is a positive integer,
must be less than or equal to
. Thus, the only integer the expression can take is
, which means the only value for
is
. Thus
~superagh
Solution 3(similiar to solution2)
Cross multiplying and adding the fraction we get the fraction to be equal to , This value has to be an integer. This implies,
.
=>
but, hence
but
does not divides
,
-(1)
=>
but,
-(2)
from (1) and (2) we get that n=42. Comparing this with the options, we see that option E is the incorrect statement and hence E is the answer.
~rudolf1279
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 10 Problems and Solutions |
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.