Question

find the smallest positive integer n , where n not equal to 13, such that the highest common factor of n-13 and 3n+8 is greater then 1??

Answer 1

Given :  highest common factor of n-13 and 3n+8 is greater than 1

To find : smallest positive integer n,

Solution:

Let say k is the  highest common factor HCF

n - 13  =  ak        

3n+ 8  =  bk    

let say n = 13 + x

=> x  = ak

3(13 + x) + 8  = bk

=> 3x + 47  = bk

=> 3ak + 47 = bk

=> 47 = k(b - 3a)

=> k  = 47   as k > 1       => k = 47      

b - 3a = 1

b = 4 , a = 1

n - 13  =  47  => n = 60      

3n+8  =  bk    => 3(60)+ 8  = 168 =  47 * 4

47 is the common factor

n = 60

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