LCM for 5, 7 and 21
What's the Least Common Multiple (LCM) of 5, 7 and 21?
Answer
(One hundred five)
105 is the smallest positive integer that 5, 7 and 21 all divide without a remainder — every other common multiple of the three is a multiple of it.
Finding the LCM of 5, 7 and 21 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 5: 5 (exponent form: 5)
All Prime Factors of 7: 7 (exponent form: 7)
All Prime Factors of 21: 3 × 7 (exponent form: 3 × 7)
| Prime factor | 5 | 7 | 21 | Highest power |
|---|---|---|---|---|
| 3 | — | — | 3 | 3 |
| 5 | 5 | — | — | 5 |
| 7 | — | 7 | 7 | 7 |
LCM = 3 × 5 × 7 = 105
Checking the LCM of 5, 7 and 21 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(5, 7) = (5 × 7) ÷ 1 = 35 ÷ 1 = 35
LCM(35, 21) = (35 × 21) ÷ 7 = 735 ÷ 7 = 105
LCM(5, 7, 21) = 105
The pairs may be taken in any order — 7 and 21 first, then 5 — and the answer is the same 105. 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