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1. Answer the following questions and justify your answer. a. Given any set S of 6...

1. Answer the following questions and justify your answer.

a. Given any set S of 6 natural numbers, must there be two numbers in S that have the same remainder when divided by 8?

b. Given any set S of 10 natural numbers, must there be two numbers in S that have the same remainder when divided by 8? 


c. Let S be a finite set of natural numbers. How big does S have to be at least so that it contains at least 5 numbers that have the same remainder when divided by 8? 


  

Math   Statistics-And-Probability   
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Answer #1

a)The remainder of a natural number when divided by 8 is the set . So tere are 8 possible remainders. You can select up to 8 natural numbers and still those have 8 different remainders. So it is not necessary that two numbers in S ( set of 6 natural numbers) that have the same remainder when divided by 8.

b) From part (a) if S contains more than 8 elements then it is necessary that two numbers in S ( set of 10 natural numbers) that have the same remainder when divided by 8.

c) Use part (a) , use Pigeonhole Principle , we can select upto 8\times 4=32 different numbers that have a maximum 4 numbers whose remainder is same. When you select the 33rd number, one of the 8 remainders in occurs more than 5 times. So the set S should contain at least 33 elements.

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