Given any set of 53 integers, show that there are two of them having the property that either their sum or their difference is evenly divisible by 103. (this is a pigeonhole principle problem).

See proof below.

Step-by-step explanation:

The Pigeonhole principle states that if we place n+1 objects in n places, then one of those n places must have more than one object. In theory, this may seem a very obvious principle but some of the problems which involve this principle can be more difficult than what you'd think of.

In this case we have to prove that given ANY set of integers, there are two of them having the property that either their sum or their difference is evenly divisible by 103.

This would translate to: if we have n and m integers in this set, we'd have one pair for which 103|(n+m) or 103|(n-m). This last condition gives us the clue of using modulos for this problem.

First, we're going to choose 52 pigeonholes (since we have 53 integers). Now, we're going to label the integers with numbers from 0 to 102 depending on their congruence modulo 103.

Once we've done this, we're going to place the integers in the pigeonhole according to their congruence, the pigeonholes will be numbered (0,103), (1,102), (2,101), (3,100)... (50,53), (51,52). (I.e: If the integer is congruent to 6 modulo 103, it will be placed in the (6,97) pigeonhole).

This way any two integers that are placed in one of these pigeonholes will be divisible by 103 (either their sum or their difference).

Note that we have 52 pigeonholes and 53 integers, therefore, one of the pigeonholes will have more than one number (two at least) and that's how we are sure it will satisfy the relation that their sum or their difference is evenly divisible by 103.

f(x) is the value of the function. b is the value of the function when x equals zero or the y-coordinate of the point where the line crosses the y-axis in the coordinate plane.

step-by-step explanation:

hope this !

answer: (yeees, you need a poet.)

//blooming serenity//

beautiful in the morning

mysterious at night

wonderous nature souring

with all it's mystical might

the flower that blooms

to the fox that looms

nocturnal creatures

with various features

a blooming flower soon in sight

as a new little hatchling fears to take flight  

afraid to leave it's calming nest,

but dear little bird, mother knows best

as sure as there are stars in the sky,

life is not about who, where what, when or why-

it's the journey that follows

adventures we swallow

of   nature's deep wonders,

natural disasters and storm's thunders.

blooming serenity

life and nature, truly is heavenly.

-kerstien lehman

step-by-step explanation: