Find number of common divisors of 463050 and 2425500

Strategy: Every common divisor is a divisor of the greatest common divisor, so we must find the greatest common divisor, then determine how many factors it has.

1. Use the Euclidean Algorithm to find the greatest common divisor.
2. Factor the greatest common divisor into primes.
3. If the greatest common divisor has prime factorization $$p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_n^{\alpha_n}$$ then a common divisor has factorization $$p_1^{\beta_1}p_2^{\beta_2} \ldots p_n^{\beta_n}$$ where $0 \leq \beta_k \leq \alpha_k$ for $1 \leq k \leq n$, so the number of divisors of the greatest common divisor is $$(\alpha_1 + 1)(\alpha_2 + 1) \ldots (\alpha_n + 1)$$

Among all the common divisors of the two numbers there is a greatest one, and this can be calculated easily: $$\gcd(463050, 2425500)=22050$$ The prime factorisation of this number is $$22050=2^13^25^27^2$$ so the number of common divisors is $$\tau(22050)=(1+1)(2+1)(2+1)(2+1)=54$$ where $\tau(n)$ is the number-of-divisors function.