Is n/18 an integer?

[Guest DSmitty]

I believe this CAT test question has the wrong answer.

Is n/18 an integer?

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

The answer key states that the answer is "both statements together are sufficient, but neither statement alone is sufficient".

The answer should be "statement 1 alone is sufficient".

The problem is titled Integer Intrigue.

Please advise if I'm mistaken. There is an identical question in the Number Properties guide on page 77. Thanks.

[mail2jkd]

This is a problem that made me think as I am trying to go through the number properties guide as well...

Here is what I think...

A. 5n/18 is a integer - the prime factors in 5 ~ 5 and the prime factors for 18 ~ 2,3,3. There is no intersection of prime factors between 5 and 18, or in other words 5 cannot be used to create 18...closer we have is 3x5=15 or 4x5=20...So for any number n, n/18 integer, 5n/18 is also an integer. Hence A alone is sufficient

B. 3n/18 is an integer. - the prime factors in 3 ~ 3 and the prime factors for 18 ~ 2,3,3. There is an intersection of 3 between both prime factors or in other words, 3 can be used to get to 18. 3x6=18 where n=6. 3n/18 is an integer, but n/18 is not. However for all other n values that satisfy 3n/18=integer...n/18 also must be an integer...There is one odd ball...hence statement b alone can't say it with certainty that n/18 is an integer.

[rfernandez]

(1) It's not true that for all values n, n/18 would be an integer. Suppose n/18 were equal to 1/5. In that case, 5n/18 would be 1 (an integer) while n/18 would not be an integer. If, instead, you chose n to be 18, then both 5n/18 and n/18 would be integers. Insufficient.

(2) Can be shown to be insufficient by a similar line of reasoning.

Check out the solution page from your CAT for a complete and in-depth solution to this problem. I guess you assumed that n must be an integer Remember that on the GMAT all variables can take on any real values unless noted otherwise.

[vgh101]

Just wanted to add that this question is also a perfect candidate for using the "FIZ" testing method discussed in the Yes/No Data Sufficiency Lab. DSmitty, you should check that lab out to nail these kinds of questions in the future.

Also, the Manhattan GMAT people are pretty sharp. You can trust their answers. :D

[esledge]

Thanks for the love, vgh101 ;-) You are right that number listing works nicely here, as it does on many Number Properties questions.

You can follow the lab example of running through FIZ (Fraction, Integer, Zero) possibilities to come up with values, but you can also generate your list another way. Start with the statements, which in this case say that "something is an integer." It's really easy to generate a list of integers with no skipped values, making it less likely that you will make assumptions and/or accidentally "cherry-pick" values that give a particular answer.

Is n/18 an integer?

(1) INSUFFICIENT:
5n/18 is an integer, so 5n/18 = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, etc. Now, we care about n/18 (that's the question) so we derive n/18 by dividing 5n/18 by 5. Thus, divide each number on the list by 5.

n/18 = 5n/18 divided by 5 = -2/5, -1/5, 0, 1/5, 2/5, 3/5, 4/5, 1, 6/5, 7/5, etc.
Is n/18 an integer? Could be an integer, could be a fraction.
(Note that we see the pattern this way: Every integer is a possible value for n/18, but the fifths between the integers are possibilities, too.)

(2) INSUFFICIENT:
3n/18 is an integer, so 3n/18 = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, etc.

n/18 = 3n/18 divided by 3 = -2/3, -1/3, 0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, etc.
Is n/18 an integer? Could be an integer, could be a fraction.
(Note that we see the pattern this way: Every integer is a possible value for n/18, but the thirds between the integers are possibilities, too.)

Now, you'll be really glad you listed numbers on any problem where you have to combine the statements; most of your work is already done. The only valid values are those that appear on both lists! The only time the fifths and third from the above lists coincide is at the integer values.

(1)&(2) SUFFICENT:

n/18 = -1, -2, 0, 1, 2, 3, etc.
Is n/18 an integer? Yes, definitely.

[j]

I understand how A and B are not sufficient on their own but not certain how C is the correct answer if you have fraction and integers in both statements alone and together there are multiple fraction and numbers that are equal. Can I get clarity on how 5n/18 and 3n/18 together let you know if n/18 is an integer...I stil don't see how it is or is not an integer.

[j]

nevermind my previous post. the numbers that coincide with statement 1 and 2 are integers so together the answer is Yes. i get it now.

[StaceyKoprince]

glad you figured it out yourself - that's always the best way! :)

[beaver82]

I want to make sure that I am understanding this correctly. Is it the first line -1,0,1,2,3,4, of (A) that should correspond with the first line of (B) -1,0,1,2,3,4 to show that n/18 is an integer. Or are we comparing the second line of (A) 2/5, etc with the (B) 1/3 to determine whether N/18 is an integer.

Thank you.

[JonathanSchneider]

The second line. The first line is where we are simply testing various integers, as we are told that 5n/18 or 3n/18 is equal to an integer. In the second line we have figured out the possible values for n/18 (the value we're after). When we look at C, we want to consider the cases where we have similar possible values from 1 and 2, aka the places where these sets overlap. They ONLY overlap on integer values, so n/18 MUST be an integer.

[guy29]

I don't believe that I agree with the logic of the analysis of this question at all. I believe the answer is B.

Here's my thinking:

3n/18 is an integer, in other words, if z is this mystery integer, then 3n/18=z . Then:

3n=z18, which is the same as n=z18/3, which can be rewritten as n=z6; since z is an integer, and any integer times any other integer must always produce an integer, statement 2 should be sufficient on its own.

The answer key must be incorrect.

[guy29]

Obviously I answered if n is an integer, not if n/18 is one: my mistake. Here's the full solution:

3n/18 is an integer, in other words, if z is some mystery integer, then 3n/18=z . Then,

3n=z18, which is the same as n/18=z/3

and the first one would follow as such: 5n/18 = y(some integer), n=18y/5 and n/18=y/5

so together n/18=z/3=y/5, then you can think of it this way: 5n/90 = z30/90 = 18y/90, and it follows that 5n = z30 = y18, which implies that since z and y are integers, 5n/30 and 5n/18 must be integers. Which means that the numerators prime factors must include at least 2,3^3, and 5. The multiples of the denominator are 2,3^3 and 5. Which means that the numerator is a multiple of the denominator, and the simplified form, n/18, is always an integer.

Not a simple one to figure out.

[esledge]

Not a simple one to figure out.

The lengthy discussion of this one indicates that others agree! Glad you worked it out.

[ahistegt]

Would it be correct if concluded the following:

Stmt 1: 5n/18 integer => n can or cannot be a multiple of 18; If n were integer, the stmt would be suff. Insuff.
Stmt 2: 3n/18=n/6 is integer. All this is telling us that n is multiple of 6 and is an integer. Insuff.

Stmt 1&2: 5n/18 and n/6 are integer only when n is a multiple of 18. Hence, answer should be C.

[JonathanSchneider]

nice work.