2006 AMC 10A Problems/Problem 20
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.