Don't know whether it is right or not ? You should check the answer !!
- If you take a number say 1, then other numbers can be 5,9,13,17,21............ as the diference between these number and 1 is a multiple of 4.
- Now, if you take 2, then other numbers can be 6,10,14,18,22........... as again difference is a multiple of 4.
Take an example of 5 distinct integers => 102 , 2014 , 517, 343 , 1009
- Here, difference of 1009 - 517 = 492 which is offcourse a multiple of 4
Take another example => 555 , 1 , 960 , 1549 ,1968
- Here, difference of 1968 - 960 = 1008 which is also a multiple of 4.
Take one more example => 129 , 2015 , 17, 740 , 1796
- Difference between 1796 - 740 is a multiple of 4.
So, if something is divisible by 4, then remainder is 0 and if not, then remainder is either 1,2,3 .
Now, I guess Dirichlet principle can be applied here also known as pigeonhole principle,
You have 5 integer numbers (Pigeons) and 4 labelled blocks (Pigeonholes) .
Now, take any 5 numbers and see their last digits and divide them by 4 and place them in the particular block , it will surely fall in the blocks labelled as 0 , 1 , 2 , 3 ( These blocks are made based on the remainder obtained by dividing the number by 4 )
- I took this example => 129 , 2015 , 17, 740 , 1796
- 129 drops into 1
- 2015 drops into 3
- 17 drops into 1
- 740 drops into 0
- 1796 drops into 0 .
Here, by the pigeonhole principle, there is definitely atleast one pair which falls in the same block.
Hence, the probability that there is a pair of integers whose difference is a multiple of 4 will be 1, that is you will always find a pair which belongs to same remainder block.
