Find Consecutive Multiples by Their Sum Calculator

Welcome to the free online calculator to find consecutive multiples by a given sum. For example, the sum of 3 consecutive multiples of 8 is 888, what are the multiples?

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 multiples by a given sum of consecutive multiples?

By the example at the beginning, the sum of 3 consecutive multiples of 8 is 888, what are the multiples?

Let these three consecutive numbers be x, x + 1 and x + 2.

Multiply by 8, it is 3 consecutive multiples of 8.

8x, 8(x + 1), 8(x + 2)

Because the sum of them is 888, so we can get an equation

8x + 8(x + 1) + 8(x + 2) = 888

Simplify and solve.

8x + 8(x + 1) + 8(x + 2) = 888

8x + 8x + 8 + 8x + 16 = 888

3 * 8x + 24 = 888

24x = 864

x = 36

So, 3 consecutive numbers are 36, 37 and 38. Their multiples of 8 are

36 * 8 = 288

37 * 8 = 296

38 * 8 = 304

Verify

288 + 296 + 304 = 888

Correct.

The second method is calculated by the arithmetic sequence.

Through the definition of consecutive multiples, we can know that consecutive multiples are a set of arithmetic sequences with multiples as tolerance.

For example, 4 consecutive multiples of 8 are 24, 32, 40 and 48. They are a set of arithmetic sequences and the common difference is 8.

According to the summation formula of the arithmetic sequence

S_n = n * a₁ + n * (n – 1) * d2

Among them, a₁ is the first term of the arithmetic sequence, n is the number of items, and d is a common difference.

Now back to the problem we want to solve: the sum of 3 consecutive multiples of 8 is 888, what are the multiples?

Here, n = 3, d = 8 and S_n = 888. Substitute them into the formula.

S_n = n * a₁ + n * (n – 1) * d2

888 = 3 * a₁ + 3 * (3 – 1) * 82

888 = 3a₁ + 3 * 2 * 82

888 = 3a₁ + 24

3a₁ = 864

a₁ = 288

Therefore, the first term of the arithmetic sequence is 288, and the remaining two numbers are 288 + 8 = 296 and 288 + 8 + 8 = 304.

Verify

2888 = 36

2968 = 37

3048 = 38

288 + 296 + 304 = 888

Obviously, they are consecutive multiples of 8, and their sum is 888.

Of course, there are other solutions, depending on your knowledge base. If you don’t want to calculate by hand, it is recommended to use the Find Consecutive Multiples by Their Sum Calculator on the page directly, enter the number and you can get the answer immediately.

Next, we will introduce how to use the calculator.

Using the calculator is simple, follow these 4 steps:

Next, let’s look at some examples.