How many integers x satisfy the condition that both 2x and 3x are perfect squares?
To solve this problem, we need to find the integers x that satisfy the given condition. Let's break down the problem step by step.
Step 1: Start by finding the prime factorization of the perfect squares 2x and 3x.
- The prime factorization of 2x can be written as 2 * 2 * (prime factors of x), where the exponent of 2 is even.
- The prime factorization of 3x can be written as 3 * 3 * (prime factors of x), where the exponent of 3 is even.
Step 2: Since we want both 2x and 3x to be perfect squares, the exponents of their prime factors must be even.
- For 2x, the exponent of 2 must be even, which means x must have an even exponent of 2 in its prime factorization.
- For 3x, the exponent of 3 must be even, which means x must have an even exponent of 3 in its prime factorization.
Step 3: Combining the conditions from step 2, x must have even exponents of both 2 and 3 in its prime factorization.
Step 4: Counting the possibilities for x:
- If x has an even exponent of 2 and an even exponent of 3 in its prime factorization, it means that x itself must be a perfect square.
- Since there are infinite perfect squares, we can conclude that there are infinitely many integers x that satisfy the given condition.
In summary, there are infinitely many integers x that satisfy the condition that both 2x and 3x are perfect squares.