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Suppose that n is a positive integer such that 9n - 2 and 7n + 33 are not relatively prime. What is the value of gcd(9n - 2, 7n + 33)?

 Sep 4, 2023
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9n  -  2   and   7n  +  33

 

if we try n==1, 2, 3, 4........ then we get:

9  -  2 ==7   and   7  +  33==40

18  -  2 ==16  and  14  +  33==47

27  -  2==25  and   21  +  33==54

36  -  2==34  and   28  +  33==61

 

From the above, you will see that: (7, 40),  (16, 47),  (25, 54),  and  (34,  61) are all relatively prime to each other. In other words, their GCD ==1. This pattern will continue all the way from n==1 until we get to:

 

n==173 and we get: (9*173  -  2)==1,555  and  (7*173  +  33)==1,244. So, now we 2 numbers: 1,555 ,  1,244.

 

Prime factorization of:

1244 = 2^2×311 (3 prime factors, 2 distinct)
1555 = 5×311 (2 distinct prime factors)

 

Therefore, GCD(1,244, 1555) ==311

 Sep 5, 2023

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