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Least Common Multiple - LCM: Problems with Solutions
Problem 1 sent by Maya
Find the least common multiple of 2 and 4.
Solution:
4 = 2 * 2.
4:2 = 2
4:4 = 1
So LCM of 2 and 4 is 4.
Problem 2
Find the least common multiple of 6 and 3.
Solution:
Since
6=3.2
, then
6|3
, so
LCM(6;3)=6
.
Problem 3
Find the least common multiple of 12 and 9.
Solution:
By definition, LCM(12;9) is the smallest number, which divides both 12 and 9. If we factorize them, we see that [tex]12=2^2.3[/tex] and [tex]9=3^2[/tex]. So, if
x
is the LCM of 12 and 9, then [tex]x|3^2[/tex],
x|3
and [tex]x|2^2[/tex].
x|3
follows from [tex]x|3^2[/tex], so we are left with [tex]x|3^2[/tex] and [tex]x|2^2[/tex]. Since [tex]3^2[/tex] and [tex]2^2[/tex] are coprime, [tex]x=2^2\cdot3^2=4\cdot9=36[/tex].
Problem 4
Determine the least common multiple of 7 and 3.
Solution:
Both 7 and 3 are prime, so they are also coprime. Then their LCM is their product, 21.
Problem 5
If
x
is the least common multiple of 8 and 9, find
x
.
Solution:
By factoring both numbers, we see that [tex]8=2^3[/tex], and that [tex]9=3^2[/tex]. They have no common prime factors, so they are coprime. Therefore their least common multiple is their product, 72.
Problem 6
Find the least common multiple of 7 and 12.
Solution:
We see that 7 is a prime number and is therefore coprime with any number, which is not divisible by 7. 7 does not divide 12, so 7 and 12 are coprime numbers. Therefore their LCM is their product, 84.
Problem 7
Find the least common multiple of
15
and
5
.
Solution:
Since
15=5 × 3
, the LCM of
15
and
5
is
15
.
Problem 8
Find the least common multiple of
4
and
10
.
Solution:
The first few multiples of 4 are 4,8,12,16,20,24,... and the first few multiples of 10 are 10,20,30,... The least number to appear in both sequences is 20, so it is the LCM of 4 and 10.
Problem 9
Find the least common multiple of 3 and 9.
Solution:
Since
9=3 × 3
, the LCM of 9 and 3 is 9.
Problem 10
Find the least common multiple of
15
and
6
.
Solution:
[tex]LCM(15;6)=LCM(3.5;3.2)=3.LCM(5;2)=3.5.2=30[/tex]
Problem 11
Find the least common multiple of
14
and
10
.
Solution:
[tex]LCM(14;10)=LCM(2.7;2.5)=2.LCM(7;5)=2.7.5=2.35=70[/tex]
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