For which natural numbers a fraction can be simplifedI want to know how to check for which natural numbers fraction 7 n + 19 7 n + 2 can be simplified. The correct answer is supposed to be n = 2 k − 1I do understand that the denominator and the numerator have to have a common divisor in order to simplify the fraction, but fail to see how this particular problem should be approached.

Question

For which natural numbers a fraction can be simplifedI want to know how to check for which natural numbers fraction             7      n      +      19              7      n      +      2       can be simplified. The correct answer is supposed to be   n  =  2  k  −  1I do understand that the denominator and the numerator have to have a common divisor in order to simplify the fraction, but fail to see how this particular problem should be approached.

Alannah Yang · Accepted Answer

7      n      +      19              7      n      +      2        =  1  +      17          7      n      +      2      Thus, in order for                     7        n        +        19                    7        n        +        2             to be simplified, we have that             17              7        n        +        2             can be simplified as well.This implies that   17 and   7  n  +  2 must have a common divisor that is not   1. Since   17 is prime, this implies that  7  n  +  2  ≡  0    (  mod    17  )This gives us that  n  ≡  7    (  mod    17  )Thus, the answer is that   n  =  17  k  +  7 where   k  ∈      Z  , not   2  k  −  1 as stated, so your answer is incorrect.

Answered question

Answer & Explanation

New Questions in Pre-Algebra