Math
Consecutive Integers Calculator
Type the target sum, how many integers you need, and whether they should be consecutive, consecutive even, or consecutive odd.
The calculator writes the equation in terms of the first integer x, simplifies it, solves for x, lists the integers, and checks the total. When the algebra gives a fraction, or an even number where an odd one was needed, it explains why no such integers exist. The product mode solves x(x + 1)… = P the same way.
The total the integers must add to. Negatives are fine.
2 for a pair, 3 for a triple, and so on (up to 1000 for sums, 6 for products).
Try an example
Result
Integers
—
- First integer x
- —
- Equation
- —
- Check
- —
- Middle / average
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the consecutive integers calculator solves
Consecutive integers follow one after another with no gaps: 27, 28, 29. Consecutive even integers (26, 28, 30) and consecutive odd integers (27, 29, 31) step by 2. Word problems give their sum or product and ask for the integers. The standard method names the first integer x, writes the others in terms of x, and solves one linear equation. This calculator shows that setup and solution for any number of integers.
| Problem | Equation | Solution |
|---|---|---|
| 3 consecutive integers, sum 84 | 3x + 3 = 84 | 27, 28, 29 |
| 2 consecutive integers, sum 51 | 2x + 1 = 51 | 25, 26 |
| 3 consecutive even integers, sum 36 | 3x + 6 = 36 | 10, 12, 14 |
| 4 consecutive odd integers, sum 40 | 4x + 12 = 40 | 7, 9, 11, 13 |
| 2 consecutive integers, product 3,192 | x(x + 1) = 3,192 | 56, 57 (or -57, -56) |
| 3 consecutive integers, sum 50 | 3x + 3 = 50 | none: x = 47/3 |
When there is no solution
The sum of k consecutive integers is k times their average, and the average is the middle integer (k odd) or a half-integer (k even). So three consecutive integers always add to a multiple of 3, and two consecutive integers always add to an odd number. For even or odd runs the average must itself be even or odd. When the target breaks these rules the equation gives a fraction or the wrong parity, and the calculator explains which rule failed.
Products
Product problems lead to polynomial equations: x(x + 1) = 3,192 becomes x^2 + x - 3,192 = 0 with roots 56 and -57, so both 56, 57 and -57, -56 work. The product mode finds every integer run by searching near the kth root of the product, which is where all the factors must sit.
How to use it
- Choose whether the problem gives the sum or the product.
- Enter the target, how many integers, and whether they are consecutive, consecutive even, or consecutive odd.
- Read the integers and the equation; open Show the work to see the setup, the simplification, and the check.
- If the page reports no solution, read the reason: a fractional x or a parity mismatch.
How to read the answer
The result lists the integers in increasing order; x is the first of them. The equation line shows the algebra a teacher expects, and the check line adds (or multiplies) the integers back to the target. For odd counts the middle integer equals the average of the run, which is a fast way to verify. In product mode two runs may appear, one negative; use the one the problem allows.
Common mistakes and edge cases
- Writing consecutive even integers as x, x + 1, x + 2. They step by 2: x, x + 2, x + 4 (odd integers use the same pattern).
- Forgetting the parity check: 3x + 6 = 39 gives x = 11, but 11 is odd, so there are no three consecutive even integers summing to 39.
- Reporting only x when the question asks for all the integers.
- Dropping the negative solution in product problems, or including it when the problem says positive.
- Assuming a decimal x means an arithmetic error; often it simply means the problem has no solution.
Worked examples
Three consecutive integers, sum 84
27, 28, 29.
Integers
27, 28, 29
Four consecutive odd integers, sum 40
7, 9, 11, 13.
Integers
7, 9, 11, 13
Three consecutive even integers, sum 36
10, 12, 14.
Integers
10, 12, 14
Negative sum
Five consecutive integers adding to -15.
Integers
-5, -4, -3, -2, -1
Sum of zero
Three consecutive integers: -1, 0, 1.
Integers
-1, 0, 1
No solution (fraction)
3x + 3 = 50 has no integer solution.
Integers
Error
No solution (parity)
Three consecutive even integers cannot sum to 39.
Integers
Error
Product of two consecutive integers
x(x + 1) = 3192 → 56, 57 or -57, -56.
Integers
-57, -56 or 56, 57
Product of three consecutive integers
14 × 15 × 16 = 3360.
Integers
14, 15, 16
Impossible product
No two consecutive integers multiply to 100.
Integers
Error
Frequently asked questions
How do you find three consecutive integers with a given sum?+
Call them x, x + 1, x + 2 and add: 3x + 3 = sum. Solve for x. For a sum of 84, x = 27 and the integers are 27, 28, 29. Shortcut: the middle one is the sum divided by 3.
How do you write consecutive even or odd integers?+
Both step by 2: x, x + 2, x + 4, … The difference is only the starting parity, which you check after solving.
Why does the equation sometimes give a fraction?+
Because the sum is impossible for that many integers. Three consecutive integers always add to a multiple of 3, and two consecutive integers always add to an odd number.
Can consecutive integers be negative?+
Yes. -5, -4, -3, -2, -1 are five consecutive integers with sum -15. Word problems sometimes restrict to positive integers, so check the wording.
How do you find two consecutive integers with a given product?+
Solve x(x + 1) = P, a quadratic. x is close to √P: for P = 3,192, √P ≈ 56.5, and 56 × 57 = 3,192. The negative pair -57 × -56 also works.
What is the sum of the first n consecutive integers?+
Starting from 1, the sum is n(n + 1)/2, the nth triangular number. Starting from x, it is n × x + n(n - 1)/2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026