Out on my street, there are a certain number of pigeons sitting on telephone poles. If there were exactly 2 pigeons sitting on each telephone pole, then 1 pigeon would have no space to sit on. If there were exactly 3 pigeons sitting on each telephone pole, then there would be 1 telephone pole with no pigeons on. How many pigeons were there?
To solve this problem, let's assume there are x pigeons and y telephone poles.
According to the first condition, if there were exactly 2 pigeons sitting on each telephone pole, then 1 pigeon would have no space to sit on. This can be represented by the equation:
x = 2y - 1 ...(Equation 1)
According to the second condition, if there were exactly 3 pigeons sitting on each telephone pole, then there would be 1 telephone pole with no pigeons on. This can be represented by the equation:
x = 3(y - 1) ...(Equation 2)
Now we have a system of equations with two unknowns (x and y). To find the solution, we can solve these equations simultaneously.
First, let's solve Equation 1 for x:
x = 2y - 1
Next, let's substitute this value of x into Equation 2:
2y - 1 = 3(y - 1)
Solving this equation, we get:
2y - 1 = 3y - 3
-y = -2
y = 2
Now substitute the value of y back into Equation 1 to find x:
x = 2(2) - 1
x = 3
Therefore, there are 3 pigeons in total.
So, the answer is 3 pigeons.