Problem

For every integer n prove that the fraction \displaystyle\frac{21n+4}{14n+3} cannot be reduced any further.

Solution

For every integer a and b, gcd(a,b)=gcd(a,b-a), where gcd(x,y) is the greatest common divisor of x and y. We shall apply this fact twice:

gcd(21n+4, 14n+3)=gcd(7n+1, 14n+3)=gcd(7n+1, 7n+2)=1.

Another approach:

3(14n+3)-2(21n+4)=1.

Thus, any common divisor of 14n+3 and 21n+4 divides 1!