Two consecutive numbers are pairs like (1, 2), (2, 3), (3, 4), (9, 10), or (100, 101). The LCM of any two consecutive numbers is simply their product. This guide explains the shortcut, the mathematical proof behind it, custom site-themed SVG diagrams, worked examples, and where it applies.
Two numbers are consecutive when they follow each other in order, differing by exactly 1. If the first number is n, the next one is n + 1.
Examples of consecutive number pairs:
Designed with website theme colors: Any divisor d of both n and (n+1) must divide their difference (n+1) - n = 1. Since only 1 divides 1, their GCD is strictly 1 (Coprime).
Instead of listing multiples or doing prime factorization, you can just multiply the two consecutive numbers. That's the LCM.
Here's how to use the trick in three quick steps:
Confirm the two numbers differ by exactly 1 (like 7 and 8, or 15 and 16).
Skip the long method — just compute n × (n + 1) in your head or on paper.
Divide the result by each number. Both divisions should give whole numbers.
The shortcut works because of a fundamental LCM rule for coprime numbers.
Now let's apply this rule to consecutive numbers:
Visualization of multiples of 4 (brand primary) and 5 (accent). Because GCD(4, 5) = 1, they have zero common multiples before their product 4 × 5 = 20.
Let's prove it formally, step by step.
For any positive integer n, the LCM of n and n + 1 equals n × (n + 1).
Let's see the shortcut in action with three different pairs.
| Step | Action | Result |
|---|---|---|
| 1 | Check: are 4 and 5 consecutive? | Yes (5 − 4 = 1) |
| 2 | Multiply directly | 4 × 5 = 20 |
| ✓ | Verify: 20 ÷ 4 and 20 ÷ 5 | 20 ÷ 4 = 5 ✓; 20 ÷ 5 = 4 ✓ |
Answer: LCM(4, 5) = 20.
| Step | Action | Result |
|---|---|---|
| 1 | Check: are 9 and 10 consecutive? | Yes (10 − 9 = 1) |
| 2 | Multiply directly | 9 × 10 = 90 |
| ✓ | Verify: 90 ÷ 9 and 90 ÷ 10 | 90 ÷ 9 = 10 ✓; 90 ÷ 10 = 9 ✓ |
Answer: LCM(9, 10) = 90.
| Step | Action | Result |
|---|---|---|
| 1 | Check: are 99 and 100 consecutive? | Yes (100 − 99 = 1) |
| 2 | Multiply directly | 99 × 100 = 9900 |
| ✓ | Verify: 9900 ÷ 99 and 9900 ÷ 100 | 9900 ÷ 99 = 100 ✓; 9900 ÷ 100 = 99 ✓ |
Answer: LCM(99, 100) = 9900.
Here are the LCM values for common consecutive number pairs. Notice how each LCM is simply the product.
The shortcut only applies to consecutive integers (numbers differing by 1). It fails for other kinds of "consecutive" pairs.
Side-by-side comparison of consecutive series: Consecutive integers and consecutive odd numbers have GCD = 1 (Product rule applies), while consecutive even numbers have GCD = 2 (Requires halving).
| Pair Type | Example | GCD | LCM | Does Trick Work? |
|---|---|---|---|---|
| Consecutive integers | (7, 8) | 1 | 56 | ✓ Yes |
| Consecutive integers | (12, 13) | 1 | 156 | ✓ Yes |
| Consecutive even numbers | (4, 6) | 2 | 12 | ✗ No |
| Consecutive odd numbers | (5, 7) | 1 | 35 | ✓ Yes* |
| Consecutive multiples of 3 | (3, 6) | 3 | 6 | ✗ No |
| Consecutive multiples of 10 | (10, 20) | 10 | 20 | ✗ No |
*Note on consecutive odd numbers: Consecutive odd numbers like 5 and 7 are always coprime (their GCD is 1), so the trick does work for them. For example, LCM(5, 7) = 5 × 7 = 35. But consecutive even numbers are not coprime, so the trick fails for those.
The shortcut is especially useful when you need fast mental math for scheduling, packing, or any situation involving coprime numbers.
| Field | Problem | Example Using the Trick |
|---|---|---|
| Scheduling | Two events repeat every n and n+1 minutes | Every 9 and 10 minutes → LCM = 9 × 10 = 90 min |
| Mechanics | Gears with consecutive tooth counts realign | 11-tooth & 12-tooth gears → 11 × 12 = 132 turns |
| Commerce | Purchase equal quantities from consecutive pack sizes | 7-pack & 8-pack → buy 56 of each |
| Fractions | Add fractions with consecutive denominators | 1/6 + 1/7 = 13/42 (denominator = 42 = 6 × 7) |
| Music | Rhythm patterns with consecutive beats per bar | 5/8 & 6/8 time → realign every 30 beats |
One of the cleanest uses of the trick is adding or subtracting fractions with consecutive denominators.
The denominator is always just the product of the two consecutive numbers!
Even experienced students fall for these traps. Watch out for them.
| ❌ Mistake | Why it's wrong | ✅ Correct approach |
|---|---|---|
| Applying the trick to 4 and 6 | 4 and 6 are consecutive even numbers, not consecutive integers. GCD(4, 6) = 2 | Use standard methods: LCM(4, 6) = 12 |
| Applying the trick to 6 and 9 | 6 and 9 are consecutive multiples of 3, not consecutive integers. GCD(6, 9) = 3 | Use standard methods: LCM(6, 9) = 18 |
| Forgetting to check GCD first | The trick only works when GCD = 1 | Always verify the numbers are coprime before multiplying |
| Multiplying huge numbers by hand | Even though the trick works, 999 × 1000 = 999000 is easy to slip up on | Use a calculator or double-check via the LCM identity |
Quick answers to the most common questions about the LCM of two consecutive numbers.
The LCM of two consecutive numbers is always their product. For example, LCM(4, 5) = 4 × 5 = 20, and LCM(9, 10) = 9 × 10 = 90.
Two consecutive integers are always coprime (their GCD is 1). For coprime numbers, LCM(a, b) = a × b. That's why the trick works every single time.
The LCM of 1 and 2 is 2, since 1 × 2 = 2.
The LCM of 10 and 11 is 110, since 10 × 11 = 110.
The LCM of 100 and 101 is 10100, since 100 × 101 = 10100.
The trick works for consecutive odd numbers like 5 and 7 (they're coprime), but it does not work for consecutive even numbers like 4 and 6 (their GCD is 2, not 1). Always check the GCD first.
Yes. Any two consecutive integers (like n and n+1) are always coprime — their GCD is always 1. This is because the only common divisor of n and n+1 must also divide their difference, which is 1.
There is no "largest" pair — for any integer n, the pair (n, n+1) has LCM = n(n+1). As n grows, the LCM grows quadratically.
Not directly. Three consecutive numbers like 4, 5, 6 are not pairwise coprime (4 and 6 share 2). For LCM(4, 5, 6) = 60, you'd need standard methods. The trick only applies to pairs of consecutive integers.
Key terms related to consecutive numbers and the LCM shortcut.