Prove that the expression  ((12n+1) / (30n+2)) is irreducible for all positive integers 'n'. 

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Please find below the solution to the asked query:

We have expression  : $\frac{\left( 12 \mathrm{n} + 1 \right)}{\left( 30 \mathrm{n} + 2 \right)}$ , Here we can see that numerator is 12 n + 1  and that is always give a ' odd number ' at any positive integer ' n ' .

As : at  n = 1 we get

12 ( 1 ) + 1 =  12 + 1 = 13 , That is a odd number ,

At  n = 2 we get

12 ( 2 ) + 1 =  24 + 1 = 25 , That is a odd number ,

At  n = 5 we get

12 ( 5 ) + 1 =  60 + 1 = 61 , That is a odd number ,

And denominator is  =  30 n + 2  and that is always give a ' even number ' at any positive integer ' n ' .

As : at  n = 1 we get

30 ( 1 ) + 2 = 30 + 2 = 32 , That is a even number

At  n = 2 we get

30 ( 2 ) + 2 = 60 + 2 = 62 , That is a even number ,


At  n = 5 we get

30 ( 5 ) + 2 = 150 + 2 = 152 , That is a even number

Therefore,

Expression  : $\frac{\left( 12 \mathrm{n} + 1 \right)}{\left( 30 \mathrm{n} + 2 \right)}$ is irreducible .                                           ( Hence proved )

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