Let a = {1, 2, 3, ..., k}. What should be the minimum value of k such that always there are at least 2 numbers having same remainder when divided by 5?
Answers
Given :
A set
a = { 1,2,3,...,k }.
To find :
Value of k for which atleast 2 number should have same remainder with 5.
Solution:
We have given a set
a = {1,2,3,...,k}
It is said that when we divide each element of this set with 5,
We should get atleast 2 such elements with same remainder.
And the number of elements in this set should be minimum.
So, firstly
On dividing any number with 5,
The possible remainders can be 0,1,2,3 and 4.
On dividing with 1 we get 1 as remainder,
Similarly on dividing with 2 we get 2 as remainder
It goes on with 3 remainder for 3, 4 remainder for 4 and 0 remainder for 5.
After that when we go with 6 we again get remainder 1 and with 7 we again get remainder 2 and goes on.
So, concentrating o the given condition,
It is said that set should have at least 2 numbers with same remainder.
So, till 6 we get two numbers 1 and 6 having same remainder (1) and, till 7 also we get at least 2 numbers (total 4 numbers) to get same same remainder.
as 1 and 6 get same remainder as 1 and 2 and 7 get same remainder as 2.
But it is also said that the value of k should be minimum.
As 6 < 7,
So
Value of k = 6
and
Set a = {1, 2, 3, 4, 5, 6}.