You don't always have a calculator handy — during tests, in class, or mental arithmetic. Learn 6 proven mental math shortcuts with Venn diagrams and decision flowcharts. Click any section below to jump directly:
This is the easiest trick of all. If the smaller number divides the larger number evenly, then the LCM is simply the larger number.
Why? Because if a divides b, then every multiple of b is automatically a multiple of a. So the smallest common multiple must be b itself.
a | b (a divides b), then LCM(a, b) = b. No calculation needed!Two consecutive numbers (like 7 and 8, or 20 and 21) are always coprime — their only common factor is 1. That means their LCM is simply their direct product.
Two numbers are coprime if their greatest common divisor is 1. When that's the case, their LCM equals their product — no matter how large the numbers are.
Why do mental tricks work? When numbers are coprime (left), their prime sets are disjoint with zero overlap, so LCM = direct product. When they share factors (right), the overlapping GCD must be divided out once.
When the numbers are small, you can find the LCM by taking multiples of the larger number and checking if each is divisible by the smaller one.
This is often the fastest trick when you're working with numbers like 6 and 8, or 9 and 12.
This is the most versatile mental trick of all. If you can quickly spot the GCD (Greatest Common Divisor), the LCM is just a single multiplication and division away.
Finding the GCD mentally is often much easier than finding the LCM directly, because GCD divides both numbers and cannot be larger than either of them.
(a ÷ GCD) × b). This keeps the numbers small in your head!This trick is best when working with slightly larger numbers. It works by testing multiples of the larger number using mental divisibility rules.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 24 ✓ |
| 3 | Sum of digits is divisible by 3 | 123 → 1+2+3=6 ✓ |
| 4 | Last two digits form a multiple of 4 | 172 → 72 ÷ 4 = 18 ✓ |
| 5 | Ends in 0 or 5 | 125 ✓ |
| 6 | Divisible by both 2 (even) and 3 | 42 ✓ |
| 9 | Sum of digits is divisible by 9 | 729 → 7+2+9=18 ✓ |
| 10 | Ends in 0 | 90 ✓ |
Use this summary table to instantly pick the fastest trick based on your pair of numbers.
| Situation | Best Trick | Example |
|---|---|---|
| One number divides the other | Trick 1 — Take larger number | LCM(4, 12) = 12 |
| Consecutive numbers | Trick 2 — Multiply them | LCM(7, 8) = 56 |
| Both prime (or coprime) | Trick 3 — Multiply them | LCM(5, 11) = 55 |
| Small numbers | Trick 4 — Doubling method | LCM(6, 8) = 24 |
| Both share a factor | Trick 5 — GCD shortcut | LCM(12, 18) = 36 |
| Larger numbers | Trick 6 — Test multiples | LCM(14, 21) = 42 |
Follow this simple visual decision flow to quickly navigate to the right shortcut.
A visual decision path for finding the LCM mentally — from the fastest 2-second check to the general formula.
Grab a pencil or do them completely in your head! Apply the tricks you just learned, then verify with the answer table.
| Problem | Answer | Shortcut Method |
|---|---|---|
| LCM(3, 12) | 12 | 3 divides 12 → take the larger number |
| LCM(8, 9) | 72 | Consecutive integers → multiply: 8 × 9 = 72 |
| LCM(5, 13) | 65 | Both prime → multiply: 5 × 13 = 65 |
| LCM(6, 10) | 30 | Doubling: 10, 20, 30 ✓ (30 ÷ 6 = 5) |
| LCM(15, 25) | 75 | (15 ÷ 5) × 25 = 3 × 25 = 75 |
| LCM(18, 27) | 54 | (18 ÷ 9) × 27 = 2 × 27 = 54 |
| LCM(22, 33) | 66 | (22 ÷ 11) × 33 = 2 × 33 = 66 |
| LCM(9, 15) | 45 | (9 ÷ 3) × 15 = 3 × 15 = 45 |
Like any mental calculation skill, speed comes with pattern recognition. Keep these 5 rules of thumb in mind:
First check if one number divides the other — if so, the LCM is the larger one. If they're consecutive or coprime, just multiply them. Otherwise, divide one number by the GCD and multiply by the other: LCM = (a ÷ GCD) × b.
Consecutive numbers are always coprime (their GCD is 1), so their LCM is simply their direct product. For example, LCM(7, 8) = 7 × 8 = 56.
Yes! But it's best for small numbers. For larger numbers, dividing by the GCD is much faster than checking multiple multiples.
Two numbers are coprime if they share no common prime factor. Common cases: two distinct primes, any consecutive integers, or powers of different primes (like 4 = 2² and 9 = 3²).
Euclid's algorithm repeatedly takes the remainder of dividing the larger number by the smaller, until the remainder is 0. The last non-zero remainder is the GCD. For example, GCD(48, 18): 48 = 18×2 + 12; 18 = 12×1 + 6; 12 = 6×2 + 0. So GCD = 6.
Not directly unless all three are pairwise coprime. For three numbers, find the LCM of the first two, then find the LCM of that result with the third. For example, LCM(4, 6, 8): LCM(4, 6) = 12; LCM(12, 8) = 24.