if x divisible by 32 and by 12 , is x divisible by 8 ?

[hoary scarab]

question

[calm void]

what do you think

[hoary scarab]

8 = 2^3

[rough atlas]

Remember that if x is divisible by all the factors of a number, then it's divisible by that number

[hoary scarab]

32 = 2^5

8 is not a factor of 12

[dim jackal]

32 is 8*4, that’s the only thing we need to know

[dim jackal]

different denominators of the same numerator don’t need to have the same factor

18 divisible by 9 and 2, for instance

[hoary scarab]

i get why x is divisible by 8, given x divisible by 32

i dont get how x is divisible by 8, given x divisible by both 32 and 12.

32 = 22222 , how do i recreate 12 with these factors ?

[hoary scarab]

2^5

[rough atlas]

Consider this

If x is divisible by both 32 and 12, it must be divisible by 2^5 and 2^2 * 3

That means that x must at the very least be 2^5 * 3

Any number that is divisible by 32 and 12 must also be divisible by 2^5 * 3

And since 8 is a factor of this prime factorization

We know now that if a number is divisible by 32 and 12 then it is divisible by 8

[hoary scarab]

so we take the lcm of the two factors ?

and see if we can create 8 out of the lcm ?

[rough atlas]

Yes

If a number is divisible by two numbers then it's divisible by their lcm

[hoary scarab]

how would the question differ if i had to use gcd ?

[rough atlas]

And if those two numbers are coprime then their lcm is just their product

[mighty tree]

it is cuz 32 is divisible by 8

[hoary scarab]

ok, this is cool

co prime is when they only have the number 1 in common ?

when do i use gcd in such ocasions ? gcd of 32 and 12 would be 2^2*3 right ? or just 2^2 ?

[rough atlas]

Yes

[rough atlas]

Not sure gcd really comes up here

But the gcd of 32 and 12 is 4

[hoary scarab]

ok

thanks, im just trying to understand when to use each one

[wet hill]

15 is divisible by 3 and 5 but 3 and 5 are mutually (and actually) prime. it doesn't matter than 12 is not divisible by 8 the fact that a known factor of x is divisible by 8 implies that 8 is also a factor of x.

[hoary scarab]

@rough atlas so based on @wet hill , if x divisible by x1,x2,x3 then if we know that 1 out of the x's numbers is divisible by 8 means that 8 is also divisible by x ?

like sort of a tree thing

with parents and childs

whats the point of finding gcd and lcm ? where do these come in handy ?

[rough atlas]

no it doesnt

32 is divisible by 8 but 8 is not divisible by 32

[hoary scarab]

im confused