Find the sum of all positive integers c such that for some positive integers a and b
{a!⋅b!=c!}
{a+c+3}
To find the sum of all positive integers c satisfying the given condition, we first need to understand the problem and break it down step by step.
Let's begin with the equation {a!⋅b!=c!}. This equation implies that the factorial of a multiplied by the factorial of b is equal to the factorial of c. In other words, (a! * b!) = c!.
Factorial means multiplying a number by all the positive integers less than it down to 1. For example, 5! (read as "5 factorial") is calculated as 5 * 4 * 3 * 2 * 1 = 120.
Now, let's consider the equation {a+c+3}. This equation represents the sum of a, c, and 3.
To find the sum of all positive integers c satisfying the given condition, we need to identify the values of a, b, and c that satisfy the equation (a! * b!) = c!, and then calculate the sum of a, c, and 3.
Here is a step-by-step approach to find the sum:
1. Start by selecting a positive integer a.
2. For each value of a, calculate the factorial of a, denoted as a!.
3. Iterate through all positive integers b greater than or equal to a.
4. For each value of a and b, calculate the factorial of b, denoted as b!.
5. Calculate the product of a! and b!, denoted as a! * b!.
6. Check if there exists any positive integer c such that c! is equal to the product obtained in step 5. If such a value exists, update the sum by adding a, c, and 3.
7. Repeat steps 3 to 6 until all possible values of a and b are considered.
After iterating through all possible values of a and b, we will have the sum of all positive integers c satisfying the given condition.
Please note that this is a computational problem, and the exact sum will depend on the limits of a and b considered in the iteration process.