LCM for 12, 30 and 84

"Least Common Multiple of 3 Numbers" Calculator
Least Common Multiple of
,
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What's the Least Common Multiple (LCM) of 12, 30 and 84?

Answer

LCM of 12, 30 and 84 is
420

(Four hundred twenty)

420 is the smallest positive integer that 12, 30 and 84 all divide without a remainder — every other common multiple of the three is a multiple of it.

Finding the LCM of 12, 30 and 84 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 12: 2 × 2 × 3 (exponent form: 22 × 3)

All Prime Factors of 30: 2 × 3 × 5 (exponent form: 2 × 3 × 5)

All Prime Factors of 84: 2 × 2 × 3 × 7 (exponent form: 22 × 3 × 7)

Prime factor123084Highest power
22222222
33333
5—5—5
7——77

LCM = 22 × 3 × 5 × 7 = 420

Checking the LCM of 12, 30 and 84 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(12, 30) = (12 × 30) ÷ 6 = 360 ÷ 6 = 60

LCM(60, 84) = (60 × 84) ÷ 12 = 5,040 ÷ 12 = 420

LCM(12, 30, 84) = 420

The pairs may be taken in any order — 30 and 84 first, then 12 — and the answer is the same 420. That is exactly why the two routes on this page agree.

Related Calculations

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About "Least Common Multiple of 3 Numbers" Calculator

This calculator finds the Least Common Multiple (LCM) of three numbers at once, without doing it in two passes. For example, it can help you find out what's the Least Common Multiple (LCM) of 12, 30 and 84? (The answer is: 420). Enter the first number (e.g. '12'), the second (e.g. '30') and the third (e.g. '84'), then hit the 'Calculate' button.
The Least Common Multiple of three numbers is the smallest positive integer that all three of them divide without a remainder
"Least Common Multiple of 3 Numbers" Calculator
Least Common Multiple of
,
and

LCM Table

Number 1Number 2Number 3LCM
1230691,380
123070420
1230714,260
123072360
1230734,380
1230742,220
123075300
1230761,140
1230774,620
123078780
1230794,740
123080240
1230811,620
1230822,460
1230834,980
123084420
1230851,020
1230862,580
1230871,740
1230881,320
1230895,340
123090180
1230915,460
1230921,380
1230931,860
1230942,820
1230951,140
123096480
1230975,820
1230982,940

FAQ

What's the Least Common Multiple (LCM) of 12, 30 and 84?

LCM of 12, 30 and 84 is 420

How to find the LCM of 12, 30 and 84 by prime factorization?

Take every prime that appears in 12, 30 and 84 to the highest power it reaches in any one of them and multiply: 2² × 3 × 5 × 7 = 420.

Can you get the LCM of 12, 30 and 84 two numbers at a time?

Yes, and it is the check this page prints. The LCM of 12 and 30 is 60, and the LCM of 60 and 84 is 420 — the same answer, because the least common multiple is associative and does not depend on the order of the three numbers.

Is 420 divisible by 12, 30 and 84?

Yes, and by definition it is the smallest positive integer that is: 420 ÷ 12, 420 ÷ 30 and 420 ÷ 84 all come out whole. Any other common multiple of 12, 30 and 84 is a multiple of 420.

See Also