Math prob-1???

Question

Prove that the fraction (21n+4)/(14n+3) cannot be reduced for any n.

Answer 1

Proof by contradiction.

if it can be reduced then
21n+4 = ab
14n+3 = ac
for some integers a,b,c not equal to 1.
42n + 8 = 2ab
42n + 9 = 3ac
1 = 3ac - 2ab
1 = a(3c - 2b)
so if a,b,c are integers then a must equal 1.
This is a contradiction, so the original fraction cannot be reduced for integer n.

Answer 2

Let there is k integer such that k= (21n+4)/(14n+3) , k>0 because (21n+4)/(14n+3)>0 .
then 21n+4=14kn+3k
21n-14kn=3k-4
7n(3-2k)=(3k-4)
n=(3k-4)/7(3-2k) but n>0
then it must be (3k-4)/7(3-2k)>0
i.e 4/3

Answer 3

one of the more simple way is
21*14/3*4 must be a right digit( not a fraction number)
294/12
24.5 is not a right digit
so can not reduced for any n