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Least Common Multiple - LCM

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Least Common Multiple - LCM: Difficult Problems with Solutions

Problem 1

Find the least common multiple of 21 and 6.

Solution:
The multiples of 6 are [tex]6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...[/tex] The multiples of 21 are [tex]21, 42, 63, ...[/tex]. We see that the smallest number, which is found in both series is 42. Therefore it is the LCM of 21 and 6.

Problem 2

Find the least common multiple of 60 and 315.

Solution:
We factorize both numbers: [tex]60=2^2 \times 3 \times 5[/tex], [tex]315=3^2 \times 5 \times 7[/tex]. We take the highest exponent of every factor, and the least common multiple is [tex]2^2 \times 3^2 \times 5 \times 7=1260[/tex].

Problem 3

Find the least common multiple of 9, 27 and 45.

Solution:
[tex]LCM(9;27;45)=LCM(9 \times 1;9 \times 3;9 \times 5)=9 \times LCM(1;3;5)=9 \times LCM(3;5)=9 \times 3 \times 5=135[/tex]

Problem 4

Find the least common multiple of 35 and 14.

Solution:
[tex]LCM(35;14)=LCM(7 \times 5;7 \times 2)=7 \times LCM(5;2)=7 \times 5 \times 2=70[/tex]

Problem 5

Find the least common multiple of [tex]15[/tex], [tex]20[/tex] and [tex]25[/tex].

Solution:
LCM(15;20;25)=LCM(5.3;5.4;5.5)=5.LCM(3;4;5)=5.3.4.5=15.20=300

Problem 6

Find the least common multiple of [tex]8[/tex], [tex]12[/tex] and [tex]20[/tex]

Solution:
[tex]LCM(8;12;20)=LCM(4 \times 2;4 \times 3;4 \times 5)=4 \times LCM(2;3;5)=4\cdot2\cdot3\cdot5=4\cdot3\cdot 2 \cdot 5=12 \cdot 10=120[/tex]

Problem 7

Find [tex]LCM(12;7;5)[/tex].

Solution:
LCM(12;7;5)=LCM(LCM(12;7);5)=LCM(12.7;5)=12.7.5=420, since 12 and 7 are coprime and so are (12.7) and 5. (In this case, it is said that the numbers are pairwise coprime)

Problem 8

Determine the least common multiple of [tex]360[/tex] and [tex]108[/tex].

Solution:
[tex]360=2^3 \cdot 3^2 \cdot 5^1[/tex]
[tex]108=2^2 \cdot 3^3 \cdot 5^0[/tex] We take the highest exponents for each base and get: [tex]LCM(360;180)=2^3 \cdot 3^3 \cdot 5^1=6^3 \cdot 5=1080[/tex]

Problem 9

Find the least common multiple of [tex]18[/tex] and [tex]45[/tex].

Solution:
LCM(18;45)=LCM(9.2;9.5)=9.LCM(2;5)=9.2.5=90

Problem 10

Find the least common multiple of [tex]2[/tex], [tex]3[/tex], [tex]4[/tex].

Solution:
[tex]LCM(2;3;4)=LCM(LCM(2;3);4)=LCM(6;4)=LCM(2 \times 3;2 \times 2)=2 \times LCM(3;2)=2 \times 6=12[/tex]

Problem 11

Determine the least common multiple of [tex]333[/tex] and [tex]74[/tex]

Solution:
LCM(333;74)=LCM(37.9;37.2)=37.LCM(9;2)=37.18=666

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