remaider problem

[clock60]

hi guys
i need some clarification, for deeper understanding the concepts of LCM and GCF, if possible please share your ideas to below

The integers k and n are such that 4 < k < n and k is not a factor of n. If R is the remainder when n is divided by k, is R > 2?
1.Thre greatest common factor of k and n is 4.
2.The least common multiple of k and n is 84.
oa is A

[aleph777]

Hi,

STMT 1: GCF of K and N is 4.
There's a simple rule that comes in handy when it comes to factors: the GCF of two numbers cannot be greater than the difference between those two numbers. In this case the GCF of K and N must be less than G - N.

That's because they're multiples of that number. So in this case, since they're both greater than 4, the next possible multiples of 4 are K = 8 and N = 12. But they could also be 8 and 20 or 12 and 28, etc. The only stipulation in the question is that K is NOT a factor of N, meaning, N can not be evenly divisible by K. So lets try any of those numbers:

12/8 = 1 r 4
20/8 = 2 r 4
28/12 = 2 r 4

Therefore: Sufficient.

STMT 2: The LCM of K and N is 84.

Let's break up 84 into all it's factors: 2, 2, 3, 7, which means we have a lot of options when breaking these factors into two numbers. K = 2 x 2 x 3 = 12 and N = 3 x 7 x 2 = 42 works fine. But so does K = 2 x 3 =6 and N = 2 x 2 x 7 = 28.

Therefore: Insufficient.

Answer: A
[spoiler][/spoiler]

[clock60]

hi aleph777, thank you for reply, i`d like to clarify one moment
i agree that 1 st sufficient, but following you rule
k=2^2*3=12
n=2^2*11=44 here GCF (k,n)=GCF(12,44)=4, and k is not factor of n and k<n
44=12*3+8
and 4<(44-12)
what i am missing?

[aleph777]

Clock60

Typo! Let me fix that right away. The rule is that the difference between two numbers with a common factor cannot be less than that factor. So the GCF of K and N MUST BE <= K - N.

And that's because, as multiples of x, those numbers are always separated by a factor of x. In this case it's 4, so, 8, 12, 16, 20, 24, etc.

Sorry for the confusion!

[clock60]

i hope you mean greater or equal, again in the example GCF(8,12)=4 and the difference between 12-8=4?