Question

Numbers are drawn from the 27 integers 1 to 27. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose difference is divisible by 4?

Please help me to solve this one😩🙏🙏

Answer 1

I experimented with a list of integers 1-27. For example, if we choose 1,2,3,4......then we'd have to jump to 9,10,11,12 to ensure no differences of 4.....then to 17,18,19,20.....and lastly to 25,26,27. That is 15 numbers altogether. Any other number drawn after that would produce a difference of 4 with one of these, so we would have to draw at least 16 numbers.
This was the max number I could find after trying other combinations as well.

Answer 2

Because two numbers whose difference is divisible by 4, we can sort the numbers from 1 - 27 like this
1 5 9 13 ... 25
2 6 10 14 ... 26
3 7 11 15 ... 27
4 8 12 16 ...
Now if we draw 2 random numbers from a random line, the difference of the 2 numbers we have drawn will be divisible by 4.
Because of that, we can draw 4 random numbers from 4 different lines
The 5th will be the last number to ensure there are two numbers whose difference is divisible by 4.
So the final answer is 5 numbers