Problem Solving

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is?

a. 6
b.12
c.24
d.36
e.48


my approach:
start with E (largest possible number) n=48 and n^2=4224 this is not divisible by 72 throw it away.
move to d, n=36 and 36^2 =1296 = 18 yes here is my answer. But problem is qa is far different than my approach.














qa is b

When dealing with multiplication always think of prime factors. n^2 is divisible by 72 and 72 =- 2*2*2*3*3.
now if n is divisible by s, then n^2 is divisible by s^2. but here 72 is not a perfect square, multiplying 72 by 2 we get 144. now it is a valid assumption as any divider for n^2 should be a perfect square.
so we end up taking sqrt 144, which gives us the answer 12...

simplyjat wrote:When dealing with multiplication always think of prime factors. n^2 is divisible by 72 and 72 =- 2*2*2*3*3.
now if n is divisible by s, then n^2 is divisible by s^2. but here 72 is not a perfect square, multiplying 72 by 2 we get 144. now it is a valid assumption as any divider for n^2 should be a perfect square.
so we end up taking sqrt 144, which gives us the answer 12...

Hi ,

Can you please further elaborate on the part in bold? Not sure I completely follow

simplyjat, I need to side with harvard dreamin because I lost you there. I dont follow you on the reasoning to the bold faced. can you help?

Enginpasa1 wrote:simplyjat, I need to side with harvard dreamin because I lost you there. I dont follow you on the reasoning to the bold faced. can you help?

I think another way of looking at the problem is this:

if a # is divided by 72, then that number can be

72*1=72
72*2=144
72*3=216
72*4=288
72*5=360

*
*
*
and so on

We are given that the number being divided by 72 is a perfect square (n^2). From the list that I've created above, you can see that 144 is a perfect square (12^2). So n = 12. The largest number that can be divided into 12 is 12.

Hope this helps

Sonia

Could you please justify why it is n cannot be 36 . As per question , n ^ 2 shud be divisible its not just n = 12 but for any value like 36 etc too can be considered right ? please clarify my doubt !

Thanks
Senthil

Enginpasa1 wrote:If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is?

a. 6
b.12
c.24
d.36
e.48


my approach:
start with E (largest possible number) n=48 and n^2=4224 this is not
divisible by 72 throw it away.
move to d, n=36 and 36^2 =1296 = 18 yes here is my answer. But problem is qa is far different than my approach.














qa is b

Ok the reason It is not divisible by 24, 36 and so on is because the questions asks you for the largest number(that means between 6 and 12).
Follow the explanation and you will understand.

Since 72 is a factor of n^2.
or in other words n^2 has factors 2,3,4,6,8,12,18,24,36 and 72
and its given that n is a positive integer. The lowest number possible for n^2 to be divisible by 72 is 72 but 72(its not a perfect square) and hence we get that n is not an integer which contradicts the question. This means n^2> 72.
The next number divisible by 72 is 144(12^2) which is also a perfect square.

36*36 = 1296. If you say "n" should be divisible by 36,(the question asks you the largest number that can divide n. What you are doing is dividing by 36*36 =1296/72 which is wrong. You are assuming n=36 that is not the question and I dont know what you are doing putting n=36 and then calculating n^2 36*36 Please read the question carefully and understand what it asks) then n*n should be divisible by 36*36 or 36^2 should be a factor of n^2 and hence n^2 must be divisible by 1296 but it may or may not be true as question say its divisible by 72 and not 1296 similar is the case for 24 and other numbers.

Reason we took 12 and not 6 is the question asks for the largest possible which is 12

precious rain explained that number divided by 72 is a perfect square, yet the question constraints never explained this.

I am getting a better handle on the question but neeed to check one thing. The reason why I didnt see the ssimplicity of the question is because 72 converting to 144. How can I fix this for future purposes. It seems that the only REAL difference between og easy and hard questions is to rephrase to curtail to your needs. TO me this is becoming one of the biggest keys to the GMAT!

I second Senthil. There is only one condition. n^2 is a multiple of 72.
So 48 ^2 is 2304 a multiple of 72 (2304/72 = 32). The highest number which can divide 48 is 48 itself. So why cannot the answer be 48.
Can someone help me please?

Thanks
~R

Enginpasa1 wrote:If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is?

a. 6
b.12
c.24
d.36
e.48

Let's start by making sure that we understand the question.

We know that:

n is a positive integer; and

n^2 is a multiple of 72.

(Since n is an integer, n^2 is an integer squared which is, by definition, a perfect square).

The question is asking us to find the largest positive integer that MUST be a factor of n.

Much of the confused replies seem to have missed the word "MUST" in the question. We're not looking for a number that COULD be a factor of n - we want one that HAS TO BE a factor of n.

In order to find numbers that MUST be factors of n, we want to make n as small as we possibly can. So, let's find the smallest perfect square that's a multiple of 72.

72, 144...

hey! There we go. So, the smallest perfect square that's a multiple of 72 is 144.

So, n^2 = 144 and, accordingly, n=12.

Therefore, the largest integer that MUST be a factor of n is 12: choose (B).

Moral of the story: make sure you understand exactly what the question is asking, or you're going to spend a lot of time looking for the wrong answer.

Stuart,

og p. 169

Thank you but i have some more questions. I can see how you moved the 72 to a 144 in order to get a perfect square of 12. BUT, lets say we use choice c. 24^2 =576 which is divisible by 72 and 144. FUrther, it is larger than 12. And 24 is larger than 12. Now what?

Hi,

Just to understand the logic of the answer, let's proceed in this way:

a) 6 - Factors are : 1,2,3,6
b) 12 - Factors are : 1,2,3,4,6,12
c) 24 - Factors are : 1,2,3,4,6,8,12,24
d) 36 - Factors are : 1,2,3,4,6,9,12,18,36
e) 48 - Factors are : 1,2,3,4,6,8,12,16,24,48

The question say n^2 divisible by 72, so A is out. We are left with B to E.

Now, it says that the largest +ve integer that must divide n?

n can be any of these nos 12, 24, 36 and 48 but in all the n which is the largest common factor : 12.

That's it.

The quickest method to solve the problem is mentioned by Stuart.

Now it clicks. Thanks folks! whew! Once in a while I get stumped like that!