Least Common Multiple - LCM: Problems with Solutions

Solution:
4 = 2 * 2.
4:2 = 2
4:4 = 1
So LCM of 2 and 4 is 4.

Solution:
Since $6=3\times2$, then [tex]6|3[/tex], so [tex]\text{LCM}(6;3)=6[/tex].

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 \times 3[/tex] and [tex]9=3^2[/tex]. So, if [tex]x[/tex] is the LCM of 12 and 9, then [tex]x|3^2[/tex], [tex]x|3[/tex] and [tex]x|2^2[/tex]. [tex]x|3[/tex] 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].

Solution:
Both 7 and 3 are prime, so they are also coprime. Then their LCM is their product, 21.

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.

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.

Solution:
Since [tex]15=5 × 3[/tex], the LCM of [tex]15[/tex] and [tex]5[/tex] is [tex]15[/tex].

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.

Solution:
Since [tex]9=3 × 3[/tex], the LCM of 9 and 3 is 9.

Solution:
[tex]LCM(15;6)=LCM(3 \times 5;3 \times 2)=3 \times LCM(5;2)=3 \times 5 \times 2=30[/tex]

Solution:
[tex]LCM(14;10)=LCM(2 \times 7;2 \times 5)=2 \times LCM(7;5)=2 \times 7 \times 5=2 \times 35=70[/tex]