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)?
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
