Question

Consecutive odd integers are odd integers listed in order such as \(5,7,\) and 9 . Find three consecutive odd integers whose sum is 111 . Are there three consecutive odd integers whose sum is \(1111 ?\) Explain.

Short answer

The three consecutive odd integers that sum up to 111 are 35, 37 and 39. As for the sum of 1111, there are no three consecutive odd integers that add up to the sum since the equation yields a decimal number.

Step-by-step solution

State the problem as an equation

Let's let the first integer be \(x\). Because odd integers are two units apart, the next consecutive odd numbers will be \(x + 2\) and \(x + 4\). For these three consecutive numbers to add to 111, we can write the equation: \(x + (x + 2) + (x + 4) = 111\)

Solve the equation

Solving the equation \(x + (x + 2) + (x + 4) = 111\), we can simplify this to \(3x + 6 = 111\). Further simplifying, we subtract 6 from both sides, we get \(3x = 105\). Dividing by 3, we have \(x = 35\). Therefore, the three consecutive odd integers are 35, 37, and 39.

Checking if there are three consecutive odd integers totaling 1111

Let's use the same principle to find if there are three consecutive odd integers that add up to 1111 by setting the equation \(x + (x + 2) + (x + 4) = 1111\). Simplifying, we get \(3x + 6 = 1111\), or \(3x = 1105\), and \(x = 368.33\). However, this is not an integer. Hence, there are no three consecutive odd integers that sum up to 1111.