Find Consecutive Multiples by Their Sum of Squares Calculator

Welcome to the free online calculator to find consecutive multiples by a given sum of squares. For example, the sum of squares of two consecutive multiples of 7 is 1225, what are the consecutive multiples?

Find Consecutive Multiples by Their Sum of Squares Calculator

The sum of squares of a consecutive multiples of b is c

Eg, the sum of squares of three consecutive multiples of 5 is 725, what are the consecutive multiples? Enter 3 in the first input box, enter 5 in the second input box and enter 725 in the third input box, then click Calculate button.

Calculate Reset

How to find the consecutive multiples by their sum of squares?

What is a multiple? If an integer a is divisible by b, then a is a multiple of b.

Consecutive multiples are consecutive numbers multiplied by the same multiple.

For example, the consecutive numbers 3, 4, 5 and 6 multiplied by 8 are 24, 32, 40 and 48. So, 24, 32, 40 and 48 are consecutive multiples of 8.

Now, how to calculate the consecutive multiples by a given sum of squares of consecutive multiples?

By the example at the beginning, the sum of squares of two consecutive multiples of 7 is 1225, what are the consecutive multiples?

Let these two consecutive numbers be x and x + 1.

Multiply by 7, it is two consecutive multiples of 7.

7x, 7(x + 1)

Because the sum of squares of them is 1225, so we can get an equation

(7x)² + [7(x + 1)]² = 1225

Simplify and solve.

(7x)² + [7(x + 1)]² = 1225

49x² + (7x + 7)² = 1225

49x² + 49x² + 98x + 49 = 1225

98x² + 98x – 1176 = 0

x² + x – 12 = 0

(x – 3) (x + 4) = 0

x = 3 or x = -4

So, two consecutive numbers are (3, 4) or (-4, -3). Their multiples of 7 are (21, 28) or (-28, -21).

Verify

21² + 28² = 441 + 784 = 1225

Or

(-21)² + (-28)² = 441 + 784 = 1225

Correct.

The process is not complicated, but requires some calculations. If you don’t want to calculate by hand, it is recommended to use the Find Consecutive Multiples by Their Sum of Squares Calculator on the page directly, enter the number and you can get the answer immediately.

Next, we will introduce how to use the calculator.

How to use the calculator to find the consecutive multiples

Using the calculator is simple, follow these 4 steps:

1Enter the number of consecutive multiples in the first input box.
2Enter the multiple in the second input box.
3Enter the sum of squares in the third input box.
4Click Calculate button to get the answer, or click Reset button to start a new calculation.

Next, let’s look at some examples.

Solved examples using find consecutive multiples by the sum of squares of consecutive multiples calculator

Example 1: The sum of squares of 3 consecutive multiples of 5 is 725, what are the consecutive multiples?

Enter 3 in the first input box.

Enter 5 in the second input box.

Enter 725 in the third input box.

Then click Calculate button, as shown in the figure, these 3 consecutive multiples (10, 15, 20) or (-20, -15, -10).

10² + 15² + 20² = 100 + 225 + 400 = 725

Or

(-20)² + (-15)² + (-10)² = 400 + 225 + 100 = 725

The sum of squares of 3 consecutive multiples of 5 is 725, what are the consecutive multiples?

Example 2: The sum of squares of two consecutive multiples of 4 is 976, what are the consecutive multiples?

Enter 2 in the first input box.

Enter 4 in the second input box.

Enter 976 in the third input box.

Then click Calculate button, as shown in the figure, these two consecutive multiples (20, 24) or (-24, -20).

20² + 24² = 400 + 576 = 976

Or

(-24)² + (-20)² = 576 + 400 = 976

The sum of squares of two consecutive multiples of 4 is 976, what are the consecutive multiples?

Find Consecutive Multiples by Their Sum of Squares Calculator