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Numbers - PigeonHole

For COMPETITION
Number of Total Problems: 2.
FOR PRINT ::: (Book)

Problem Num : 1

From : NCTM

Type: None

Section:Numbers

Theme:None
Adjustment# :

Difficulty: 1

Category PigeonHole

Analysis

Solution/Answer

Problem Num : 2

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
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?
$mathrm{(A) } frac{1}{2}qquadmathrm{(B) } frac{3}{5}qquadmathrm{(C) } frac{2}{3}qquadmathrm{(D) } frac{4}{5...$
'

Category PigeonHole

Analysis

Solution/Answer
For two numbers to have a difference that is a multiple of 5, the numbers must be congruent $mod{5}$ (their remainders after division by 5 must be the same).
$0, 1, 2, 3, 4$ are the possible values of numbers in $mod{5}$ . Since there are only 5 possible values in $mod{5}$ and we are picking $6$ numbers, by the Pigeonhole Principle, two of the numbers must be congruent $mod{5}$ .
Therefore the probability that some pair of the 6 integers has a difference that is a multiple of 5 is $1 Longrightarrow mathrm{E}$ .

Answer:

Array ( [0] => 5975 [1] => 7839 ) 2