Question

Six distinct positive integers are randomly chosen between $1$ and $2006,$ inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5?$

$\dfrac{1}{2}$

$\dfrac{3}{5}$

$\dfrac{2}{3}$

$\dfrac{4}{5}$

$1$

Answer: E

Concepts:pigeonhole principle modular arithmetic

Difficulty rating: 1510

Solution:

Group the integers by their remainder modulo $5.$ There are only $5$ possible remainders but $6$ integers, so by the Pigeonhole Principle two share a remainder.

Their difference is then a multiple of $5.$ This always happens, so the probability is $1.$

Thus, the correct answer is E.

2006 AMC 10A Problem 20

Answer: E

Difficulty rating: 1510

Solution:

Problem 20 in Other Years