I don't even know where to start with this. -_-
As A and B range over all ordered pairs of distinct coprime positive integers, how many different possibilities are there for:
gcd((A+B)^12, (A-B)^61)
To find the number of different possibilities for gcd((A+B)^12, (A-B)^61), we need to consider all possible values of A and B, where A and B are distinct coprime positive integers.
Let's break down the problem step by step:
Step 1: Find the prime factorization of (A+B)^12 and (A-B)^61.
- Start by expanding the expressions using binomial expansion, and then write them as a product of primes raised to certain powers.
- For example, let's say (A+B)^12 = p1^a1 * p2^a2 * p3^a3 * ... and (A-B)^61 = q1^b1 * q2^b2 * q3^b3 * ..., where p1, p2, p3, ..., q1, q2, q3, ... are distinct primes.
Step 2: Determine the common factors.
- Find the common prime factors between (A+B)^12 and (A-B)^61 by comparing their prime factorizations.
- The common factors will be the primes that appear in both factorizations with the same or lower exponent.
Step 3: Find the number of possibilities.
- Count the number of different possibilities by considering the different combinations of common prime factors that can occur.
- For example, if there are k common prime factors, you can have 2^k different possibilities since each prime can either be included or excluded.
So, to get the answer to the question, you need to calculate the prime factorization of (A+B)^12 and (A-B)^61, determine the common prime factors, and count the number of different possibilities based on the common factors.
Note: This explanation provides a general approach to solving the problem. Implementing this approach in practice might involve considering specific properties of exponents and prime factorizations for simplification.