I want to know how to check for which natural numbers fraction (7n+19)/(7n+2) can be simplified. The correct answer is supposed to be n=2k−1. I 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.

Jacoby Erickson

Jacoby Erickson

Answered question

2022-10-11

For which natural numbers a fraction can be simplifed
I 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 1
I 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.

Answer & Explanation

Alannah Yang

Alannah Yang

Beginner2022-10-12Added 22 answers

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.

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