Difference between revisions of "2006 AMC 10A Problems/Problem 20"
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== Problem == | == Problem == | ||
− | Six distinct | + | Six distinct positive integers are randomly chosen between <math>1</math> and <math>2006</math>, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of <math>5</math>? |
− | <math>\ | + | <math>\textbf{(A) } \frac{1}{2}\qquad\textbf{(B) } \frac{3}{5}\qquad\textbf{(C) } \frac{2}{3}\qquad\textbf{(D) } \frac{4}{5}\qquad\textbf{(E) } 1\qquad</math> |
+ | |||
== Solution == | == Solution == | ||
− | For two numbers to have a difference that is a multiple of 5, the numbers must be congruent <math>\bmod{5}</math> (their | + | For two numbers to have a difference that is a multiple of <math>5</math>, the numbers must be congruent <math>\bmod{5}</math> (their remainders after division by <math>5</math> must be the same). |
+ | |||
+ | <math>0, 1, 2, 3, 4 </math> are the possible values of numbers in <math>\bmod{5}</math>. Since there are only <math>5</math> possible values in <math>\bmod{5}</math> and we are picking <math>6</math> numbers, by the [[Pigeonhole Principle]], two of the numbers must be congruent <math>\bmod{5}</math>. | ||
+ | |||
+ | Therefore the probability that some pair of the <math>6</math> integers has a difference that is a multiple of <math>5</math> is <math>\boxed{\textbf{(E) }1}</math>. | ||
− | + | ==Video Solution== | |
+ | https://youtu.be/jfkW_KwI9Wo | ||
− | + | ~savannahsolver | |
== See also == | == See also == |
Latest revision as of 08:21, 17 March 2023
Contents
Problem
Six distinct positive integers are randomly chosen between and
, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of
?
Solution
For two numbers to have a difference that is a multiple of , the numbers must be congruent
(their remainders after division by
must be the same).
are the possible values of numbers in
. Since there are only
possible values in
and we are picking
numbers, by the Pigeonhole Principle, two of the numbers must be congruent
.
Therefore the probability that some pair of the integers has a difference that is a multiple of
is
.
Video Solution
~savannahsolver
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 |
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