Question

Find the smallest positive integer n, where n is not equal to 13, such that the highest common factor of n-13 and 3n-8 is greater than 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 + 31  = bk

=> 3ak + 31 = bk

=> 31 = k(b - 3a)

=> k  = 31   as k > 1       => k = 31  

b - 3a = 1

b = 4 , a = 1

n - 13  =  31  => n = 44        

3n- 8  =  bk    => 3(44) - 8  = 124 =  31 * 4

31 is the common factor

n = 44    

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