Find the sum of all positive integers m such that 2^m can be expressed as sums of four factorials (of positive integers).
To find the sum of all positive integers m such that 2^m can be expressed as sums of four factorials, we can first write an equation using the properties of factorials.
Let's start by observing that the factorial of any positive integer is always divisible by 2. Therefore, all four factorials in the sum must also be divisible by 2.
The smallest factorial that is divisible by 2 is 2! = 2. So, we can start by assuming that all four factorials are at least 2. Therefore, our equation becomes:
2^m = 2 + 2 + 2 + 2
Simplifying this equation, we get:
2^m = 8
To solve for m, we can take the logarithm base 2 of both sides of the equation:
log₂(2^m) = log₂(8)
m * log₂(2) = log₂(8)
Since log₂(2) is equal to 1, we have:
m = log₂(8)
Using a calculator, we find that log₂(8) is equal to 3.
So, the sum of all positive integers m such that 2^m can be expressed as sums of four factorials is 3.