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Is the least common multiple of an even $2k$ and odd number $2l+1$ always the product of both numbers $2k(2l+1)$ ?

And also is the least common multiples of two odd numbers the product of both odd numbers?

Thank you.

Peter
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PhysicsEnthu
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    No on both. Try a bunch of examples... – coffeemath Apr 13 '19 at 11:53
  • If the difference between the odd and the even number is $1$ , or between the odd numbers is $2$ , then both results hold in general. Otherwise, there are plenty of counterexamples. In fact $lcm(a,b)=ab$ is true if and only if $\gcd(a,b)=1$ – Peter Apr 13 '19 at 11:58
  • Their LCM is their product $\iff$ they're coprime $\iff \gcd(k,2l+1) = 1\ \ $ – Bill Dubuque Apr 13 '19 at 13:47

4 Answers4

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For two numbers $a,b$ we have $$lcm(a,b)=\frac{ab}{\gcd(a,b)}$$ so even if $a$ is odd and $b$ is even , the result need not be $ab$. The same situation when both numbers are odd.

Peter
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No, the least common multiple is the least number which is divisible by both the numbers. Counter Examples

$$1)LCM(6,9)=18\neq 54 = 6\times 9$$ $$2)LCM(3,9)=9\neq 27 = 3\times 9$$

Hope it is helpful:)

Martund
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If we take 6 and 15 then their lcm is 30 so your first question has answer "no"

If we take 3 and 15 then their lcm is 15 so your second question has answer "no"

user587756
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No. The only times either works, is if gcd(x,y)=1 . Otherwise, it fails 15 and 3 for example have gcd of 3. that mean's if we multiply them to get their product, we have a number with a factor of 9 which neither number has on it's own. We have to only have factors the original numbers have.

This works in general, lcm(4,4)=4 not 16, lcm(6,3)=6 not 18, lcm(8,12)=24 not 96, etc.