Let's start by setting up the problem. We know that the least integer in the set is -25 and the sum of the integers is 26. We can represent this set of consecutive integers with an arithmetic sequence.
Let's assume the number of integers in the set is 'n'.
The sum of an arithmetic sequence can be calculated using the formula:
Sum = (n/2) * (first term + last term)
In this case, the first term is -25, and the last term can be found by adding (n - 1) to the first term.
Substituting these values into the formula, we have:
26 = (n/2) * (-25 + (-25 + (n - 1)))
Now, let's solve for 'n':
26 = (n/2) * (-25 + (-25 + n - 1))
26 = (n/2) * (-25 + n - 26)
26 = (n/2) * (n - 51)
52 = n^2 - 51n
n^2 - 51n - 52 = 0
Now, we can solve this quadratic equation for 'n' by factoring or using the quadratic formula.
Factoring:
(n - 52)(n + 1) = 0
Setting each factor equal to zero, we can find the possible values of 'n':
n - 52 = 0 or n + 1 = 0
n = 52 or n = -1
Since the number of integers cannot be negative, we discard the solution 'n = -1'.
Therefore, there are 52 integers in the set.