# Find 3 Consecutive Odd Integers Whose Sum is 219 by 3 Ways

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The sum of three consecutive odd integers is 219, what are these odd integers? I can easily tell you that the answer are 71, 73 and 75. You must be interested in how to find these 3 consecutive odd integers whose sum is 219. There are three methods here, let us introduce them one by one below.

## 1. Find three consecutive odd integers whose sum is 219 by hypothetical method

Assuming that 2 * N + 1 is used to represent the first odd integer, so the second odd integer is 2 * N + 3, the third odd integer is 2 * N + 5.

Therefore, the sum of 3 consecutive odd integers is 219, which can be expressed by the equation

(2 * N + 1) + (2 * N + 3) + (2 * N + 5) = 219

Solve this equation

(2 * N + 1) + (2 * N + 3) + (2 * N + 5) = 219

2 * N + 1 + 2 * N + 3 + 2 * N + 5 = 219

6 * N + 1 + 3 + 5 = 219

6 * N + 9 = 219

6 * N = 219 – 9

6 * N = 210

N = 210 / 6

N = 35

So, 2 * N + 1 = 2 * 35 + 1 = 71

Now, we can get that 71 is the first integer of 3 consecutive odd integers whose sum is 219. So, the second odd integer is 73, the third odd integer is 75.

The answer came out, the sum of 3 consecutive odd integers is 219, these three odd integers are 71, 73 and 75.

Verify: 71 + 73 + 75 = 219. Correct!

This is the most common method, let’s look at the second method, which is my favorite method.

## 2. Find three consecutive odd integers whose sum is 219 by formula method

According to the consecutive integers calculator based on sum, we can know that the sum of M consecutive odd integers is S, the first odd integer formula is

First(o) = S / M – M + 1

M represents the number of consecutive odd integers.

S stands for sum.

Back to the problem we want to solve: find three consecutive odd integers whose sum is 219. In here, M = 3, S = 219. Replace them in the formula to calculate the first odd integer

First(o) = 219 / 3 – 3 + 1 = 73 – 3 + 1 = 71

The first odd integer is 71, so, it can be easily calculated that the second odd integer is 73, the third odd integer is 75.

Thus, the answer are also 71, 73 and 75. Same as the first method. Then look at the third method, which is the simplest one.

## 3. Find three consecutive odd integers whose sum is 219 by average method

The principle is very simple, because we want to calculate the consecutive odd integers, so after calculating the average, we can find the odd integers near the average.

The sum of 3 consecutive odd integers is 219, so, the average of these 3 consecutive odd integers is 219 / 3 = 73. The odd integers around 73 are 69, 70, 71, 72, 73, 74, 75, 76, 77. Now the problem is simple, find 3 consecutive odd integers from 69 to 77 and their average is 73. The answer are 71, 73 and 75.

So the sum of 3 consecutive odd integers is 219, these odd integers are 71, 73, 75. The results are consistent with the above two methods. Is it very simple?

## Problems can be sloved by this answer

Now, we have found out these 3 consecutive odd integers whose sum is 219, some problems can be easily solved.

• The sum of three consecutive odd integers is 219, the smallest one is 71.
• The sum of three consecutive odd integers is 219, the middle odd integer is 73.
• The sum of three consecutive odd integers is 219, the greatest odd number is 75.
• The sum of three consecutive odd integers is 219, the average of these odd integers is 73.
• The sum of three consecutive odd integers is 219, the product of them is 71 * 73 * 75 = 388725.
• The sum of three consecutive odd integers is 219, the sum of their squares is 712 + 732 + 752 = 15995.

## Summarize

On this page, In addition to introducing three methods how to find three consecutive odd integers whose sum is 219, it also provides a calculator that calculates three consecutive odd integers based on the sum. If you encounter a similar problem next time, you can directly use this calculator to calculate the answer, which is very convenient.

Of course, if the problem you encounter is more complicated, such as: the number of consecutive integers is not 3, or you need to calculate consecutive integers or consecutive even integers. You can use our other more advanced sum-based consecutive integers calculator, where you can specify the number of consecutive integers and select consecutive integers type: natural integers, odd integers or even integers. I believe it can help you.

Well, that’s it, the above are three solutions to find three consecutive odd integers whose sum is 219. Personally, I prefer the second method, because it can accurately calculate the first integer and the calculation process is very simple. So how about you? Please leave a message and tell me which method you like?