# What two consecutive integers sum of squares is 113?

The sum of squares of two consecutive integers is 113. What are these two consecutive numbers? The answer is 7 and 8. You must be interested in how to find these two consecutive integers whose sum of squares is 113. Here will introduce a calculator and 2 methods to solve this problem. The calculator is suitable for everyone, including laymen. Two methods are suitable for those who have some foundation in mathematics. let’s start.

## 1. Sum-Squares-Based 2 Consecutive Integers Calculator

Result:
Verify:

With the help of the above calculator, we can easily find 2 consecutive integers whose sum of squares is 113. The specific steps are as follows:

• 1Enter 113 into the input box.
• 2Click the calculation button.

In the blink of an eye, the answer will appear. As shown in the figure, there are two sets of answers, one set is 7 and 8, and the other set is -8 and -7.

Very simple, if you have other similar problems: find 2 consecutive integers based on the sum of squares. It can be calculated by the sum-squares-based 2 consecutive integers calculator.

## 2. Hypothetical method

Assuming that N is used to represent the first integer, then the second integer can be represented by N + 1. Now, the sum of squares of 2 consecutive integers is 113, which can be expressed by the equation

N2 + (N + 1)2 = 113

This is a quadratic equation in one variable. When we solve this equation, we can get the value of the first integer N.

N2 + (N + 1)2 = 113

N2 + N2 + 2 * N + 1 = 113

2 * N2 + 2 * N + 1 = 113

2 * N2 + 2 * N + 1 – 113 = 0

2 * N2 + 2 * N – 112 = 0

N2 + N – 56 = 0

(N – 7) * (N + 8) = 0

N = 7 or N = -8

So, the value of the first integer is 7 or -8, then the second integer is N + 1 = 8 or -7.

Obviously, there are 2 sets of answers for which the sum of the squares of two consecutive numbers is 113. The positive integers are 7 and 8, the negative integers are -8 and -7. It is consistent with the answer calculated by the calculator in the first method!

## 3. Average method

1
##### Calculate the average of the sum of squares first.

That is, divide the sum of squares by the number. Here it is 113 / 2 = 56.5.

2
##### Then calculate the square root of the average.

Here it is √56.5 = 7.5.

3
##### Finally, find consecutive integers near the square root value.

In here, find 2 consecutive integers around 7.5, and their average value is equal to 7.5.

Through the above 3 steps, you can find that these 2 consecutive integers are 7 and 8.

Let us verify that the answer is correct?

72 + 82 = 49 + 64 = 113

Next, we consider the negative forms of these two integers -7 and -8, and we can get another answer.

## Problems can be solved by two consecutive integers whose sum of squares is 113

Now, we have found 2 consecutive integers whose sum of squares is 113. At the same time, the following problems can also be solved incidentally.

• 1The sum of squares of two consecutive integers is 113, and their sum is 7 + 8 = 15.
• 2The sum of squares of two consecutive integers is 113, and their product is 7 * 8 = 56.
• 3The sum of squares of two consecutive integers is 113, and the sum of their cubes is 73 + 8 3 = 855.
• 4The sum of squares of two consecutive integers is 113, and the smaller positive integer is 7.
• 5The sum of squares of two consecutive integers is 113, and the larger positive integer is 8.
• 6The sum of squares of two consecutive integers is 113, and their average is 7.5.

Of course, in addition to the three methods described above for finding two consecutive integers whose sum of squares is 113, there are other methods. Have you encountered them? If so, please leave a message to tell us, thank you!

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