LCM for 2, 7 and 12
What's the Least Common Multiple (LCM) of 2, 7 and 12?
Answer
(Eighty-four)
84 is the smallest positive integer that 2, 7 and 12 all divide without a remainder — every other common multiple of the three is a multiple of it.
Finding the LCM of 2, 7 and 12 by Prime Factorization
Split all three numbers into prime factors, then take every prime that shows up in at least one of them, each raised to the highest power it reaches there. Multiplying those highest powers gives the least common multiple:
All Prime Factors of 2: 2 (exponent form: 2)
All Prime Factors of 7: 7 (exponent form: 7)
All Prime Factors of 12: 2 × 2 × 3 (exponent form: 22 × 3)
| Prime factor | 2 | 7 | 12 | Highest power |
|---|---|---|---|---|
| 2 | 2 | — | 22 | 22 |
| 3 | — | — | 3 | 3 |
| 7 | — | 7 | — | 7 |
LCM = 22 × 3 × 7 = 84
Checking the LCM of 2, 7 and 12 Two Numbers at a Time
The least common multiple is associative, so three numbers can be handled as two pairs: first the LCM of the first two, then the LCM of that value and the third. Each pair goes through the two-number formula, which uses the greatest common divisor:
LCM(a, b) = (a × b) ÷ GCD(a, b)
LCM(2, 7) = (2 × 7) ÷ 1 = 14 ÷ 1 = 14
LCM(14, 12) = (14 × 12) ÷ 2 = 168 ÷ 2 = 84
LCM(2, 7, 12) = 84
The pairs may be taken in any order — 7 and 12 first, then 2 — and the answer is the same 84. That is exactly why the two routes on this page agree.
Related Calculations
See Also
- Greatest Common Factor of 3 Numbers - Find the Greatest Common Factor (GCF) of three numbers
- Least Common Multiple - Find the Least Common Multiple (LCM) of two numbers
- Greatest Common Factor - Find the Greatest Common Factor (GCF) of two numbers
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LCM Table
About "Least Common Multiple of 3 Numbers" Calculator
The Least Common Multiple of three numbers is the smallest positive integer that all three of them divide without a remainder