Is n/18 an integer? (1) 5n/18 is an integer. (2) 3n/18 is an integer.

(1) If 5n/18 is an integer, call it k, we know that n/18=k/5, which may or may not be an integer. NOT SUFF

(2) Similarly, if 3n/18 is an integer, call it m, we know that n/18= m/3, which may or may not be an integer NOT SUFF


(1) and (2): For some integers k and m, k/5=m/3 => 3k=5m =>k=5m/3.

Thus m is a multiple of 3, so n/18=m/3 is an integer SUFF

lets see..the stem says nothing about N being an integer...soo

1) 5 N/18 - integer...well N can be an integer multiple of 18 or it can be 18M/5 where M is an integer not divisible by 5...Insuff

2) simplify this to N/6, so N has prime factors 2 and 3, we still dont know if it has enuff 3s or not...but we noe for sure that N is not a fraction...

combining them we now now that N is not a fraction and that its a mutliple of 18....

C it is...

Answer woud be A if we are told in the stem that N is an integer..

From F.S 1, we know that \(\frac{5n}{18}\) is an integer. For\(\frac{n}{18} = 1\), we have a YES. Again, for \(\frac{n}{18} = \frac{1}{5}\) , we have a NO.Insufficient.

From F.S 2, we know that\(\frac{3n}{18}\) is an integer. For \(\frac{n}{18} = 1\), we have a YES,but for \(\frac{n}{18} = \frac{1}{3}\) , we have a NO.Insufficient.

Taking both together, we know that from F.S 1, either\(\frac{n}{18} = \frac{k}{5}\) or \(\frac{n}{18} = p\) , where k,p are integers and k and 5 are co-primes.

But, for \(\frac{n}{18} = \frac{k}{5}\), \(\frac{3*n}{18} = \frac{3*k}{5}\) and it will not be an integer. Thus, \(\frac{n}{18} = p\) can the only be form possible.

C.

We need to find whether n is a multiple of 18 or n= 18I for some integer I

St 1: 5n/18= Integer or n = 18*Integer/ 5
Now if Integer = 2, then n =7.2 and n/18 is not an integer

but integer = 5 then n = 18 and 18/18 is an integer

------Possible values of n = 3.6,7.2,10.8,14.4,18......

So A and D ruled out

St 2: we see that n is a multiple of 6 so possible values of n =6,12,18......
If n= 6 then n/18 is not an integer but if n=18 then n/18 is an integer.
So option B ruled out

Combining we get possible values of n =18,36,54 and so on

Hence Ans C

It's C, but I think this is as clear as it can be explained:

S1: \(\frac{5n}{18}=K1\) (Letting K1 be an integer)

Then, isolating n we get \(n=\frac{18K1}{5}\)

Going back to the original question if n is divisible by 18, based on this answer is "Maybe" (provided that K1 is a multiple of 5)

Insufficient.


S2: \(\frac{3n}{18}=K2\) (Letting K2 be an integer)

Then, isolating n we get \(n=\frac{18K2}{3}\), reducing \(n=6K2\)

Going back to the original question if n is divisible by 18, based on this answer is "Maybe" (Provided that K2 is a multiple of 3)

Insufficient.


S1 & S2: We will make the expressions for n equal:

\(\frac{18K1}{5}=6K2\), simplifying
\(K2=\frac{3K1}{5}\)
The key here is understanding that K1 and K2 MUST be integers. As such, the "Maybes" of S1 and S2 are proven to be true.
From the expression above, K2 is a multiple of 3 and K1 is a multiple of 5.

This is perhaps an extended explanation, but I think is clear enough.

Statement I is insufficient:

n = 36 (YES)
n = 18/5 (NO)

Statement II is insufficient:

n = 36 (YES)
n = 6 (NO)

Combining is sufficient:

Since 5 and 3 are co primes to each other n will have to be a multiple of 18 for 5n/18 and 3n/18 to be an integer.

Answer is C

Umm...I am having 1 query ...very simple

I don't want to get into such complex solution my only query is ...

S 1 : tells 5n/18 is an integer - 5 is prime, hence it must be clear that "n" will be divisible by 18 , in that case n/18 will be an integer .

then why some folks are not mentioning A wrong ??

Is i am missing any logic plz explain ?

You are missing that we are NOT told that n is an integer. So, 5n/18=integer, does not necessarily mean that n is a multiple of 18, if for example n=18/5, then it's not so.

Hope it's clear.

I am absolutely stumped on understanding the above explanations. My thought process was to break down the denominator into its primes --> 3^2 and 2, then identify whether N has those same characteristics, leading my to choose A.

I know this is a vague response, but any chance you could help me understand how A is wrong and how the 2 combined MUST form an integer.

Than you very much for giving me an insight...i will take care this in future.

If the qustion says division result (n/18) is an integer (no fractional part) , shoul we consider decimal values also?

Hi SoumiyaGoutham,

Integers by definition are Non-Fractional and Non-Decimal type Numbers. So (5n/18) should be an Integer doesn't ascertain the value of n which theoretically can be a fraction of two integers (or Decimal value) as well e.g. for n= 18/5, (5n/18) will be an Integer

"An integer (from the Latin integer meaning "whole")[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and √2 are not.

The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,[1] and their additive inverses (the negative integers, i.e. −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold \mathbb{Z} (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[2][3] ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers."

Ref: https://en.wikipedia.org/wiki/Integer

Statement 1: 5n is a multiple of 18 or n is a multiple of 3.6. Not sufficient
Statement 2: 3n is a multiple of 18 or n is a multiple of 6. Not sufficient

Both 1 and 2. n is a multiple of both 3.6 and 6 i.e, n is a multiple of 18. Sufficient

Hence C.

niks18 pushpitkc pikolo2510 amanvermagmat

Any idea why Bunuel never took in to account why prime factorization of 18.

(1) (5n) / (2*3^2)
(2) (3n) / (2*3^2)

I did so but still could not combine statements properly and selected E instead of C

Hi adkikani

We want to know whether \(\frac{n}{18}=Integer =>\) whether \(n=18I\) (for some integer \(I\)).

This implies two things A) whether \(n\) is integer? & B) if \(n\) is an integer then whether it is a multiple of \(18\)?

Statement 1: \(\frac{5n}{18} = Integer => n=\frac{18k}{5}\) (for some integer \(k\)), if \(k\) is a multiple of \(5\), then \(n\) is an integer and a multiple of \(18\) and if \(k\) is not a multiple of \(5\), then it is not. Insufficient

Statement 2: \(\frac{3n}{18}=Integer => n=6q\) (for some integer \(q\)). This CONFIRMS that \(n\) is an INTEGER. but we do not know yet that \(n\) is a multiple of \(18\). Insufficient

Combining 1 & 2: because \(n\) has to be an integer so we have, \(n=\frac{18k}{5}=Integer\). This is only possible if \(k\) is a multiple of \(5\). Hence \(n\) will be a multiple of \(18\). Sufficient

Option C

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Hi there!

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Regards,
Fabio.