show that anynatural no. n the fraction 21n+4/14n+3is in its lowest term.

To prove that  is a fraction in its lowest term, all we need show is that the HCF of (21n +4) and (14n + 3) is 1.
Using Euclid’s algorithm to find the HCF of (21n +4) and (14n + 3).
Since, (21n + 4) > (14n + 3), we apply the division lemma  to (21n +4) and (14n + 3), to get
(21n +4) = 1×(14n + 3) + (7n + 1)
Since the remainder  7n + 1 is not equal to zero, we apply the  division lemma to (14n + 3) + (7n + 1), to get
(14n + 3) = 2×(7n + 1) + 1
The remainder has thus come out to be 1, which will definitely divide 7n +1. 
So the divisor at the stage where we get remainder equal to zero is 1 which is hence also the HCF of (21n +4) and (14n + 3).
Thus is a fraction in its lowest term.