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Difference between revisions of "2012 AIME II Problems/Problem 12"

(Created page with "== Problem 12 == For a positive integer <math>p</math>, define the positive integer <math>n</math> to be <math>p</math>''-safe'' if <math>n</math> differs in absolute value by mo...")
 
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== Problem 12 ==
 
== Problem 12 ==
 
For a positive integer <math>p</math>, define the positive integer <math>n</math> to be <math>p</math>''-safe'' if <math>n</math> differs in absolute value by more than <math>2</math> from all multiples of <math>p</math>. For example, the set of <math>10</math>-safe numbers is <math>\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}</math>. Find the number of positive integers less than or equal to <math>10,000</math> which are simultaneously <math>7</math>-safe, <math>11</math>-safe, and <math>13</math>-safe.
 
For a positive integer <math>p</math>, define the positive integer <math>n</math> to be <math>p</math>''-safe'' if <math>n</math> differs in absolute value by more than <math>2</math> from all multiples of <math>p</math>. For example, the set of <math>10</math>-safe numbers is <math>\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}</math>. Find the number of positive integers less than or equal to <math>10,000</math> which are simultaneously <math>7</math>-safe, <math>11</math>-safe, and <math>13</math>-safe.
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== Solution ==
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== See also ==
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{{AIME box|year=2012|n=II|num-b=11|num-a=13}}

Revision as of 17:21, 31 March 2012

Problem 12

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.


Solution

See also

2012 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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