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# number theory

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As n ranges over the positive integers, what is the maximum possible value for the greatest common divisor of 11n+3 and 6n+7?

Jun 29, 2022

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Use Euclidean's Algorithm as follows:

$$11n + 3 \div 6n+7 = 1 \ \text{remainder} \ 5n -4$$

$$6n + 7 \div 5n - 4 = 1 \ \text{remainder} \ n + 11$$

$$5n - 4 \div n + 11 = 5 \ \text{remainder} \ -59$$

$$\gcd(n + 11, 59)$$

So, the maximum value for the GCD is $$\color{brown}\boxed{59}$$, and occurs at $$n = 48$$

Jun 29, 2022