Manhatten - Number system

Question

If 12 is a factor of xyz, is 12 a factor of xy?
Can some one please explain the reasoning behind this problem?

cipher · Answer

I think there is some thing missing in the question.. Anyway I will try to answer as per my understanding of the question.

I think by xyz you mean X*Y*Z, if yes then the 12 may or may not be a factor of X*Y.

For eg. x= 3, y =4, z= 5
then x*y*z = 3*4*5 ( then 12 is a factor ) and so is x*y as well

but if x=3, y=5, z=4 then 12 is a factor of x*y*z but x*y is not ..!

hardnstrong · Answer

This looks like a DS question

Silvers · Answer

Hi this seems to be a DS question, could you post the conditions so that we can take a shot at it, at this point the answer cannot be determined.

Hussain15 · Answer

I think the requirement here is to answer in "Yes", "No" or "Can not be determined".

So in my view "Can not be determined".

dixitraghav · Answer

in diffrent condition , sometimes it is , but in other conditions it is not , perhaps some conditions are missing here in the questions..............

Pipp · Answer

12 could be a factor or x alone, y alone or z alone
12 could be the factor of all three ( x=12n y=12m z=12p)
12 could be the factor of any two of them (xy,yz,xz)
12 could be the factor of (xyz) ex: 3*2*4 where neither one of them individually is a multiple of 12
So, we cant determine the answer to the question unless we know the constraints

rdevorse · Answer

This is a problem of divisibility and primes from MGMAT practice set #1. 
The problem infers to the multiplication of xyz and xy.
The answers are "Yes", "No", or "Cannot be determined".

I am working my way through it right now and have the same question. Although I understand why it CBD (xyz are unknown integers, and could be any number with the factors of 2, 2, and 3) I don't understand the explanation given by the text:

"If xyz is divisible by 12, its prime factors include 2, 2, and 3, as indicated by the prime box to the left. Those prime factors could all be factors of x and y, in which case 12 is a factor of xy. For example, this is the case when x = 20, Y= 3, and z = 7. However, x and y could be prime or otherwise not divisible by 2, 2, and 3, in which case xy is not divisible by 12. For example, this is the case when x = 5, y = II, and z = 24."

Can someone please break this down like a fraction for me? Is it basically saying that since xy are unknowns, they can be any integer and possibly inclusive or not inclusive of the factors 2, 2, and 3?

KarishmaB · Answer

Since you already understand that answer must be CBD here, I will not explain the solution. I will only break down the explanation as per your request.

"If xyz is divisible by 12, its prime factors include 2, 2, and 3,"

If a number is divisible by 5, it means it has 5 as a factor. If a number is divisible by 6, it means it has 2 and 3 as factors. If a number 'xyz' (consider it a single number obtained by multiplying 3 numbers) is divisible by 12, it means it has 2, 2 and 3 as factors i.e. it must have two 2s and a 3. If xyz = 4*3*5, it has two 2s and a 3. If xyz = 3*5*7, it doesn't have two 2s and hence is not divisible by 12. If xyz = 2*15*7, it has only one 2 and is hence not divisible by 12. So, for xyz to be divisible by 12, the product must have two 2s and a 3.

"Those prime factors could all be factors of x and y, in which case 12 is a factor of xy."
'Those prime factors' refers to 'two 2s and a 3'. Say, if the two 2s and the 3 are present in xy itself, then it doesn't matter what z is; 12 will be a factor of xy. Say, if xy = 4*9, we already have two 2s and a 3 in there, so 12 is a factor of xy.

"However, x and y could be prime or otherwise not divisible by 2, 2, and 3, in which case xy is not divisible by 12."
If x and y are both prime say x = 2 and y = 3, it is not possible that there are two 2s and 3 in xy. You can have one 2 and a 3 but not two 2s. Also, the primes could be totally different, say, x = 5 and y = 11. In this case, 12 is not a factor of xy.
Also, it is possible that x and y are not prime but are still not divisible by 12 i.e. they still do not have two 2s and a 3. Say x = 9 and y = 10. Here, we have one 2 and one 3 but we do not have two 2s. Hence xy is not divisible by 12 in this case.

In short, it is saying what you said: 
"Since x and y are unknowns, they can be any integers and possibly inclusive or not inclusive of the factors 2, 2, and 3."

rdevorse · Answer

Karishma! Thank you so much for the detailed explanation! I appreciate your time to help me understand the basics lol. I swear sometimes I stare at this stuff for too long..

Manhatten - Number system

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