LCM for 60, 84 and 120

"Least Common Multiple of 3 Numbers" Calculator
Least Common Multiple of
,
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What's the Least Common Multiple (LCM) of 60, 84 and 120?

Answer

LCM of 60, 84 and 120 is
840

(Eight hundred forty)

840 is the smallest positive integer that 60, 84 and 120 all divide without a remainder — every other common multiple of the three is a multiple of it.

Finding the LCM of 60, 84 and 120 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 60: 2 × 2 × 3 × 5 (exponent form: 22 × 3 × 5)

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

All Prime Factors of 120: 2 × 2 × 2 × 3 × 5 (exponent form: 23 × 3 × 5)

Prime factor6084120Highest power
222222323
33333
55—55
7—7—7

LCM = 23 × 3 × 5 × 7 = 840

Checking the LCM of 60, 84 and 120 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(60, 84) = (60 × 84) ÷ 12 = 5,040 ÷ 12 = 420

LCM(420, 120) = (420 × 120) ÷ 60 = 50,400 ÷ 60 = 840

LCM(60, 84, 120) = 840

The pairs may be taken in any order — 84 and 120 first, then 60 — and the answer is the same 840. 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 60, 84 and 120? (The answer is: 840). Enter the first number (e.g. '60'), the second (e.g. '84') and the third (e.g. '120'), 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
6084105420
608410622,260
608410744,940
60841083,780
608410945,780
60841104,620
608411115,540
60841121,680
608411347,460
60841147,980
60841159,660
608411612,180
608411716,380
608411824,780
60841197,140
6084120840
608412150,820
608412225,620
608412317,220
608412413,020
608412510,500
60841261,260
608412753,340
608412813,440
608412918,060
60841305,460
608413155,020
60841324,620
60841337,980
608413428,140

FAQ

What's the Least Common Multiple (LCM) of 60, 84 and 120?

LCM of 60, 84 and 120 is 840

How to find the LCM of 60, 84 and 120 by prime factorization?

Take every prime that appears in 60, 84 and 120 to the highest power it reaches in any one of them and multiply: 2³ × 3 × 5 × 7 = 840.

Can you get the LCM of 60, 84 and 120 two numbers at a time?

Yes, and it is the check this page prints. The LCM of 60 and 84 is 420, and the LCM of 420 and 120 is 840 — the same answer, because the least common multiple is associative and does not depend on the order of the three numbers.

Is 840 divisible by 60, 84 and 120?

Yes, and by definition it is the smallest positive integer that is: 840 ÷ 60, 840 ÷ 84 and 840 ÷ 120 all come out whole. Any other common multiple of 60, 84 and 120 is a multiple of 840.

See Also