If \(t\) and \(t +3\) have no factors in common, then \(t\) is not a multiple of \(3\).
Here \(t\) is a positive integer, "no factors in common" means no common factor other than \(1\).
i) Justify the original statement.
ii) Write the converse and justify it.
i) Proposition P : If \(t\) and \(t +3\) have no factors in common, then \(t\) is not a multiple of \(3\).
If suppose \(t\) is multiple of \(3\)
then \(t+3\) is
Then is a common factor
which contradicts the fact that
Therefore \(t\) is .
So, proposition is
ii) Converse: "If \(t\) then \(t\) and \(t+3\)
We know that common factor divides their differene
therefore, \(t+3 - t\ =\)
It can therefore be either \(1\) or \(3\). Since \(3\) \(t\)
The only common factor is .
So, converse is .