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For how many unique coordinate points (P, Q), such that P and Q are in

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For how many unique coordinate points (P, Q), such that P and Q are in  [#permalink]

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New post Updated on: 30 Mar 2017, 09:34
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For how many unique coordinate points (P, Q), such that P and Q are integers, is it true that P^2 - Q^2 = 1155?

a) 16
b) 24
c) 32
d) 48
e) 64

Originally posted by daboo343 on 30 Mar 2017, 09:30.
Last edited by Bunuel on 30 Mar 2017, 09:34, edited 1 time in total.
Renamed the topic and edited the question.
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For how many unique coordinate  [#permalink]

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New post 30 Mar 2017, 09:34
daboo343 wrote:
For how many unique coordinate points (P, Q), such that P and Q are integers, is it true that P2 - Q2 = 1155?

a) 16
b) 24
c) 32
d) 48
e) 64


This is a copy of the following MGMAT question: https://gmatclub.com/forum/for-how-many ... 97167.html
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Re: For how many unique coordinate points (P, Q), such that P and Q are in  [#permalink]

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New post 05 Jul 2018, 08:10
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P^2-Q^2=(P+Q)(P-Q)=1155=3*5*7*11
We can think of this as two numbers multiply with each other, and we need to find out the different combination to make the multiple of these two numbers equal to 1155.
3,5,7,11 these 4 numbers can have 8 different combination to get 1155.
(1,3*5*7*11);(3,5*7*11);(5,3*7*11);(7,3*5*11);(11,3*5*7);(3*5,7*11);(5*7,3*11);(3*7,5*11)
These 4 numbers can also be negative, so we have another 8 different combination to get 1155.
So we totally have (8+8)*2 = 32 different unique combination.
Why we need to multiple by 2?
IF A*B=1155, the value is assigned to A can also be assigned to B. For example, 15*77 and 77*15, both equal to 1155, but they are different points on coordinate.
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For how many unique coordinate points (P, Q), such that P and Q are in  [#permalink]

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New post Updated on: 08 Jul 2019, 21:27
Can I do this question by using the number of distinct factors formula?
Which says if A x B = N where N = a^p . b^q . c^r . d^s , where a,b,c,d are prime factors,
then the number of distinct factors of A x B = (p+1)(q+1)(r+1)(s+1)
which turns out to be 2.2.2.2 = 16.
Now P and Q both can take these distinct values, therefore the total number of possible values= 16 . 2 =32
The number of distinct factors will give us how many different values can A and B take.

Originally posted by baljitbagga on 08 Jul 2019, 17:05.
Last edited by baljitbagga on 08 Jul 2019, 21:27, edited 1 time in total.
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Re: For how many unique coordinate points (P, Q), such that P and Q are in  [#permalink]

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New post 08 Jul 2019, 20:20
baljitbagga wrote:
Can I do this question by using the number of distinct factors formula?
Which says if A x B = N where N = a^p . b^q . c^r . d^s , where a,b,c,d are prime factors,
then the number of distinct factors of A x B = (p+1)(q+1)(r+1)(s+1)
which turns out to be 2.2.2.2 = 32.
Number of distinct factors will give us how many different values can A and B take.


This multiplication equals to 16.

Quote:
which turns out to be 2.2.2.2 = 32.


I think you need to add 1 also as a factor. So the factors are 1,3,5,7 and 11.

Now it will be 32.

chetan2u : Please suggest.
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Re: For how many unique coordinate points (P, Q), such that P and Q are in  [#permalink]

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New post 15 Nov 2019, 05:11
How can we consider combinations od (P+Q) and (P-Q) as the possible answer choices for (P,Q)?
It is quite possible that P and Q turn out to be fractions while they add up to the possibilities. And it is clearly mentioned that P and Q are integers.
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Re: For how many unique coordinate points (P, Q), such that P and Q are in   [#permalink] 15 Nov 2019, 05:11
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