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What is the remainder

by selango » Sat Jul 03, 2010 3:20 am
If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15?

1) The remainder when p + n is divided by 5 is 1.

2) The remainder when p - n is divided by 3 is 1.

OA E

Can anyone explain in detail?
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by sanju09 » Sat Jul 03, 2010 3:43 am
selango wrote:If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15?

1) The remainder when p + n is divided by 5 is 1.

2) The remainder when p - n is divided by 3 is 1.

OA E

Can anyone explain in detail?

We can write (p^2 - n^2)/15 as [(p + n)/5] [(p - n)/3]. In any case we need to have the exact value of p^2 - n^2 in order to answer the stem, that's why each statement alone is not sufficient here.

When taken together, we get

p^2 - n^2 = (5 a + 1) (3 b + 1) form, where the quotients a and b are some positive integers, never know what.

[spoiler]E[/spoiler]
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by Patrick_GMATFix » Sat Jul 03, 2010 9:51 pm
This is a very tough GMATPrep number properties question.

Although I agree with sanju's final answer (E), I don't agree with his statement that "we need to know the exact value of p^2-n^2 in order to answer the stem".

Consider for example that for the same question, a hypothetical statement told us that "When (p^2-n^2) is divided by 45, the remainder is 12". From this we would not know the exact value of (p^2-n^2) since it could be 45+12, 90+12 or 135+12. However, we would have enough information to determine that (p^2-n^2) divided by 15 will also have a remainder of 12 (feel free to try any possible value). Therefore the contention that we need to know the exact value to determine the remainder is false


The minimum data necessary in remainder questions is unfortunately less simple. Let r be the remainder when (p^2-n^2) is divided by 15. This would mean that (p^2-n^2) = 15x + r where x is an unknown integer. For instance if the remainder of division by 15 is 3, then (p^2-n^2) could be 3 (which is 15*0+3), 18 (which is 15*1+3), 33 (which is 15*2+3)...

To determine r, we need to be able to express (p^2-n^2) in this format (the sum of a multiple of 15 and a known value). Thus, my hypothetical statement above ("When (p^2-n^2) is divided by 45, the remainder is 12") would be sufficient because it would allow us to write (p^2-n^2)=45x+12 --> (p^2-n^2)=15*3x+12 (the sum of a multiple of 15 and 12)

The correct answer to the question is indeed E, not because we don't know the value of (p^2-n^2), but because we never have enough information to express it in the format 15x + r where r is known.

A detailed solution and step-by-step video solution to this question are available at GMATPrep Question 1283

My 2 cents,
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by clock60 » Sun Jul 04, 2010 3:06 am
very hard to me, i agree that the answer is E, but want to go a little bit deeper

(p-n)(p+n)=(5a+1)(3b+1)
if the right part is always divisible by 15 then p^2-n^2 is also divisible, out task is to prove that both yes and no anwers are possible

if the number is divisible by 15, it can be 15,30,45,60 and so on....
let us check 30

(5a+1)(3b+1)=30. and for sure a,b,must be integers
30=2*15,5*6
in no cases a, or b are integers

45=3*15,9*5 here a and b are also not integers

60=6*10

5a+1=6. 5a=5, a=1.
3b+1=10, 3b=9, b=3
(5*1+1)(3*3+1)=6*10=60 divisible by 15
as we have both yes and no answers the answer to the problem is E

(to me unsolvable in test constraints)
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by sanju09 » Mon Jul 05, 2010 2:06 am
Patrick_GMATFix wrote:This is a very tough GMATPrep number properties question.

Although I agree with sanju's final answer (E), I don't agree with his statement that "we need to know the exact value of p^2-n^2 in order to answer the stem".

Consider for example that for the same question, a hypothetical statement told us that "When (p^2-n^2) is divided by 45, the remainder is 12". From this we would not know the exact value of (p^2-n^2) since it could be 45+12, 90+12 or 135+12. However, we would have enough information to determine that (p^2-n^2) divided by 15 will also have a remainder of 12 (feel free to try any possible value). Therefore the contention that we need to know the exact value to determine the remainder is false


The minimum data necessary in remainder questions is unfortunately less simple. Let r be the remainder when (p^2-n^2) is divided by 15. This would mean that (p^2-n^2) = 15x + r where x is an unknown integer. For instance if the remainder of division by 15 is 3, then (p^2-n^2) could be 3 (which is 15*0+3), 18 (which is 15*1+3), 33 (which is 15*2+3)...

To determine r, we need to be able to express (p^2-n^2) in this format (the sum of a multiple of 15 and a known value). Thus, my hypothetical statement above ("When (p^2-n^2) is divided by 45, the remainder is 12") would be sufficient because it would allow us to write (p^2-n^2)=45x+12 --> (p^2-n^2)=15*3x+12 (the sum of a multiple of 15 and 12)

The correct answer to the question is indeed E, not because we don't know the value of (p^2-n^2), but because we never have enough information to express it in the format 15x + r where r is known.

A detailed solution and step-by-step video solution to this question are available at GMATPrep Question 1283

My 2 cents,
-Patrick
Thanks for your valuable input, Patrick. I am sorry for not being able to listen to the explanation in the video solution as forwarded by you, thanks to my out of order sound system.

What minimum do we need to answer the following question?

What is the remainder when x is divided by 9?

Do we need all possible values of x, any one possible value of x, the exact value of x, or nothing?

Please guide.

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by san2009 » Mon Jul 05, 2010 4:21 am
Patrick_GMATFix wrote:This is a very tough GMATPrep number properties question.

Although I agree with sanju's final answer (E), I don't agree with his statement that "we need to know the exact value of p^2-n^2 in order to answer the stem".

Consider for example that for the same question, a hypothetical statement told us that "When (p^2-n^2) is divided by 45, the remainder is 12". From this we would not know the exact value of (p^2-n^2) since it could be 45+12, 90+12 or 135+12. However, we would have enough information to determine that (p^2-n^2) divided by 15 will also have a remainder of 12 (feel free to try any possible value). Therefore the contention that we need to know the exact value to determine the remainder is false


The minimum data necessary in remainder questions is unfortunately less simple. Let r be the remainder when (p^2-n^2) is divided by 15. This would mean that (p^2-n^2) = 15x + r where x is an unknown integer. For instance if the remainder of division by 15 is 3, then (p^2-n^2) could be 3 (which is 15*0+3), 18 (which is 15*1+3), 33 (which is 15*2+3)...

To determine r, we need to be able to express (p^2-n^2) in this format (the sum of a multiple of 15 and a known value). Thus, my hypothetical statement above ("When (p^2-n^2) is divided by 45, the remainder is 12") would be sufficient because it would allow us to write (p^2-n^2)=45x+12 --> (p^2-n^2)=15*3x+12 (the sum of a multiple of 15 and 12)

The correct answer to the question is indeed E, not because we don't know the value of (p^2-n^2), but because we never have enough information to express it in the format 15x + r where r is known.

A detailed solution and step-by-step video solution to this question are available at GMATPrep Question 1283

My 2 cents,
-Patrick
Patrick: it says at the bottom of your signature that GMAT fix drill is available with solutions for BTG members for 2 weeks...how do I access it? thanks
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by Patrick_GMATFix » Mon Jul 05, 2010 5:25 am
san2009 wrote:Patrick: it says at the bottom of your signature that GMAT fix drill is available with solutions for BTG members for 2 weeks...how do I access it? thanks
Send me a PM and I would be glad to help you :-)
sanju09 wrote:What minimum do we need to answer the following question?
What is the remainder when x is divided by 9?
A multiple of 9 is a number that can be written as 9a where a is an integer. A number that when divided by 9 has a remainder of, for example, 4, is a number that can be written as 9a + 4. Such numbers include 4 = 9*0+4, 13 = 9*1+4, and 22 = 9*2+4

Thus if we call the remainder r, we can say that our number is 9a+r. Thus my rephrase to the question "What is the remainder when x is divided by 9?" Would be "x = 9a+r, what is r?". To have sufficient data, I would need information that allows me to express x as 9a+r where a is an integer and r is known (we don't need to know the value of a).

An example of a sufficient statement would be: "When x+3 is divided by 27, the remainder is 15"

From this statement, we have no way to know what x is, but we can write x+3 =27b+15, so x=27b+12 and, to put it in the format that is useful to us, x=9(3b)+12. Thus you can see that we are able to express x as 9a+r where a is an integer and r is known. Thus the statement above is sufficient.

Too bad your speakers don't work. You can practice similar questions from GMATPrep by setting topic='Number Properties' in the drill generator.

Hope that helped,
-Patrick

NOTE: the actual value of the remainder in the hypothetical statement above is not 12 because the remainder of division by 9 cannot be greater than 8. To algebraically show the remainder we would need to transform x=9(3b)+12 to x=9(3b)+9+3 and ultimately to x=9(3b+1)+3. The remainder of division by 9 is 3.
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by sanju09 » Tue Jul 06, 2010 3:34 am
Patrick_GMATFix wrote:
san2009 wrote:Patrick: it says at the bottom of your signature that GMAT fix drill is available with solutions for BTG members for 2 weeks...how do I access it? thanks
Send me a PM and I would be glad to help you :-)
sanju09 wrote:What minimum do we need to answer the following question?
What is the remainder when x is divided by 9?
A multiple of 9 is a number that can be written as 9a where a is an integer. A number that when divided by 9 has a remainder of, for example, 4, is a number that can be written as 9a + 4. Such numbers include 4 = 9*0+4, 13 = 9*1+4, and 22 = 9*2+4

Thus if we call the remainder r, we can say that our number is 9a+r. Thus my rephrase to the question "What is the remainder when x is divided by 9?" Would be "x = 9a+r, what is r?". To have sufficient data, I would need information that allows me to express x as 9a+r where a is an integer and r is known (we don't need to know the value of a).

An example of a sufficient statement would be: "When x+3 is divided by 27, the remainder is 15"

From this statement, we have no way to know what x is, but we can write x+3 =27b+15, so x=27b+12 and, to put it in the format that is useful to us, x=9(3b)+12. Thus you can see that we are able to express x as 9a+r where a is an integer and r is known. Thus the statement above is sufficient.

Too bad your speakers don't work. You can practice similar questions from GMATPrep by setting topic='Number Properties' in the drill generator.

Hope that helped,
-Patrick

NOTE: the actual value of the remainder in the hypothetical statement above is not 12 because the remainder of division by 9 cannot be greater than 8. To algebraically show the remainder we would need to transform x=9(3b)+12 to x=9(3b)+9+3 and ultimately to x=9(3b+1)+3. The remainder of division by 9 is 3.
When r is the question itself, how can we expect it to be known so directly in any statement. Please address the fundamental question only.

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by Patrick_GMATFix » Tue Jul 06, 2010 6:29 am
Hey Sanju,

I believe I did address the fundamental question you raised. If a question asks "What is the remainder when x is divided by 9?", you will have sufficient data if you can express x as 9a+r where a is an integer and r is known.

You will of course not be given r directly. A statement would not say "When x is divided by r, the remainder is 4". However, you may very well be given the remainder of another division. I gave you such a statement: "When x+3 is divided by 27, the remainder is 15"

Often in remainder questions (especially in DS), the statements give you the remainder of another division. Another example would be "x + 12 is divisible by 9". This is also remainder data from another division; it tells us that when x+12 is divided by 9, the remainder is 0. which would translate to x+12=9b+0 --> x=9b-12 --> x=9(b-1)-3. This is sufficient since we can express x as 9a+r where a is an integer and r is known.

You can see for yourself the type of statements you get from these questions. Go to https://www.gmatfix.com/solutions_search, then enter keyword "remainder" and select question type "Data Sufficiency". When you hit search similar GMATPrep questions will come up and you will see that the statements provided typically give you remainder information from another division.

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by sanju09 » Wed Jul 07, 2010 12:23 am
Patrick_GMATFix wrote:Hey Sanju,

I believe I did address the fundamental question you raised. If a question asks "What is the remainder when x is divided by 9?", you will have sufficient data if you can express x as 9a+r where a is an integer and r is known.

You will of course not be given r directly. A statement would not say "When x is divided by r, the remainder is 4". However, you may very well be given the remainder of another division. I gave you such a statement: "When x+3 is divided by 27, the remainder is 15"

Often in remainder questions (especially in DS), the statements give you the remainder of another division. Another example would be "x + 12 is divisible by 9". This is also remainder data from another division; it tells us that when x+12 is divided by 9, the remainder is 0. which would translate to x+12=9b+0 --> x=9b-12 --> x=9(b-1)-3. This is sufficient since we can express x as 9a+r where a is an integer and r is known.

You can see for yourself the type of statements you get from these questions. Go to https://www.gmatfix.com/solutions_search, then enter keyword "remainder" and select question type "Data Sufficiency". When you hit search similar GMATPrep questions will come up and you will see that the statements provided typically give you remainder information from another division.

-Patrick
agreed, thanks Patrick B-)
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by GMATGuruNY » Wed Jul 07, 2010 7:14 pm
selango wrote:If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15?

1) The remainder when p + n is divided by 5 is 1.

2) The remainder when p - n is divided by 3 is 1.

OA E

Can anyone explain in detail?
Here's another approach. We can rewrite the question as follows:

What is the remainder when (p+n)(p-n) is divided by 15?

The two statements give us information about p+n and p-n. Let's make a list of the possible values that p+n and p-n could assume.

Statement 1:

If when p + n is divided by 5 the remainder is 1, then p + n must be a (multiple of 5) + 1. Why? So that when we divide p + n by 5, we'll have 1 left over.

p + n = 1, 6, 11, 16, 21, 26, 31... (Remember that 0 is a multiple of every number; that's why 1 is included in our list.)

But statement 1 tells us nothing about p - n. INSUFFICIENT.

Statement 2:

If when p - n is divided by 3 the remainder is 1, then p - n must be a (multiple of 3) + 1. Why? So that when we divide p - n by 3, we'll have 1 left over.

p - n = 1, 4, 7, 10, 13, 16...(Remember that 0 is a multiple of every number; that's why 1 is included in our list.)

But statement 2 tells us nothing about p + n. INSUFFICIENT.

Statements 1 and 2 together:

p + n = 1, 6, 11, 16, 21, 26, 31...
p - n = 1, 4, 7, 10, 13, 16...

If p = 11 and n = 10, then p>n and p + n = 21 and p - n = 1, both of which are included in our lists above.
In this case, (p + n)(p - n) = 21 * 1 = 21. What's the remainder when 21 is divided by 15? 21/15 = 1 R6. The remainder is 6.

If p = 16 and n = 15, then p>n and p + n = 31 and p - n = 1, both of which are included in our lists above.
In this case, (p + n)(p - n) = 31 * 1 = 31. What's the remainder when 31 is divided by 15? 31/15 = 1 R16. The remainder is 16.

Since the remainder can be different values, INSUFFICIENT.

The correct answer is E.
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by sam117 » Sat May 26, 2012 7:57 am
Patrick_GMATFix wrote:Hey Sanju,

I believe I did address the fundamental question you raised. If a question asks "What is the remainder when x is divided by 9?", you will have sufficient data if you can express x as 9a+r where a is an integer and r is known.

You will of course not be given r directly. A statement would not say "When x is divided by r, the remainder is 4". However, you may very well be given the remainder of another division. I gave you such a statement: "When x+3 is divided by 27, the remainder is 15"

Often in remainder questions (especially in DS), the statements give you the remainder of another division. Another example would be "x + 12 is divisible by 9". This is also remainder data from another division; it tells us that when x+12 is divided by 9, the remainder is 0. which would translate to x+12=9b+0 --> x=9b-12 --> x=9(b-1)-3. This is sufficient since we can express x as 9a+r where a is an integer and r is known.

You can see for yourself the type of statements you get from these questions. Go to https://www.gmatfix.com/solutions_search, then enter keyword "remainder" and select question type "Data Sufficiency". When you hit search similar GMATPrep questions will come up and you will see that the statements provided typically give you remainder information from another division.

-Patrick
In x=9(b-1)-3, r = -3. How come the remainder is negative? Should we arrange the equality so that x=9(b-2) +6 and so the real remainder is 6? Thanks
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by Patrick_GMATFix » Sat May 26, 2012 8:45 am
Hey Sam,

In my example, when I wrote x=9(b-1)-3, the remainder was not -3 (remainder is never negative). The point of my post was that because we can write x in the form 9a + r where a is an integer and r is known (it is -3), we have enough info to find the remainder of division by 9. Since we're dealing with Data Sufficiency, I didn't bother to actually find the remainder; it's enough to know that I can find it. In fact, if I wanted to find the remainder, I would get a remainder of 6 as you did. There is just no point in taking the time to find out this information in a DS question.

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