Difference between revisions of "1986 AIME Problems/Problem 5"
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In a similar manner, we can apply synthetic division. We are looking for <math>\frac{n^3 + 100}{n + 10} = n^2 - 10n - 100 - \frac{900}{n + 10}</math>. Again, <math>n + 10</math> must be a factor of <math>900 \Longrightarrow n = \boxed{890}</math>. | In a similar manner, we can apply synthetic division. We are looking for <math>\frac{n^3 + 100}{n + 10} = n^2 - 10n - 100 - \frac{900}{n + 10}</math>. Again, <math>n + 10</math> must be a factor of <math>900 \Longrightarrow n = \boxed{890}</math>. | ||
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+ | == Solution 2 == | ||
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+ | After applying long division, we see that <math>\frac{n^3+100}{n+10} = n^2 - 10n + 100 - \frac{900}{n+10}</math>. Thus, <math>n+10</math> must be a factor of <math>900</math>, and if we want the largest value of <math>n</math>, we have that <math>n+10 = 900 \Longrightarrow n = \boxed{890}</math>. | ||
== See also == | == See also == |
Revision as of 00:31, 31 January 2018
Contents
Problem
What is that largest positive integer for which is divisible by ?
Solution
If , . Using the Euclidean algorithm, we have , so must divide 900. The greatest integer for which divides 900 is 890; we can double-check manually and we find that indeed .
In a similar manner, we can apply synthetic division. We are looking for . Again, must be a factor of .
Solution 2
After applying long division, we see that . Thus, must be a factor of , and if we want the largest value of , we have that .
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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