Respuesta :
Among any group of d+1 integers there are two with exactly the same remainder when divided by d.
What are Positive Integers?
Positive integers are those bigger than 0 that don't contain any fractional portions. On the number line, these figures are located to the right of 0.
Is 0 a Positive Integer?
Since it is neither positive nor negative, 0 is not a positive integer. Positive integers are defined as numbers greater than 0 in mathematics.
The possible remainders after dividing an integer by d are 0, 1, 2,..., d-1.
The number of remainders that can be obtained is d, or |0, 1, 2,..., d-1|=d0,1,2,...,d-1=d.
According to the Pigeonhole Theory:
objects are the number of integers in d+1.
"holes" = "remainders" + "d"
[d+1/d]=2
According to the Pigeonhole Principle, there are always two d+1 integers in every group that have precisely the same residual after being divided by d.
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There are always two d+1 numbers in any group that have the exact same residual after division.
Examples of integers and what they are:
An integer, pronounced "IN-tuh-jer," is an entire number which can be positive, minus, even zero and is not a fraction. Integer examples include: -5, 1, 5, 8, 97, among 3,043. 1.43, 1 3/4, 3.14, and other values which aren't integers seem to be some examples.
Briefing:
The potential remainders after dividing an integer by d are 0, 1, 2,..., d-1.
The possible number of remainders is d, or I{0,1,2,.......d-1}I = d
By the Pigeonhole Principle:
Objects = umber of integers = d+1
Holes = number of remainders = d
[tex]\frac{d+1}{d} = 2[/tex]
According to the Pigeonhole Principle, there are always two d+1 numbers in every group that have precisely the same residual after being divided by d.
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