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

This is a copy of the following MGMAT question: https://gmatclub.com/forum/for-how-many ... 97167.html
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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.

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.

This multiplication equals to 16.



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.

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.

This is a tough one! The first thing you should do is that we should factor the difference of squares P2 − Q2 into (P − Q)(P + Q). That gives us:

(P − Q)(P + Q) = 1155

That means that (P - Q) and (P + Q) must both be factors of 1155, since the product of (P − Q) and (P + Q) equals 1155. Because the product of (P − Q) and (P + Q) = 1155, if we can figure out the total number of factors 1155 has, we can figure out the number of possible values for P and Q, because every unique pair of values for (P − Q) and (P + Q) will correspond to a unique value pair for (P,Q).

To find the total number of factors in 1155, we need to take its prime factorization. 1155 ends in a 5, so we can take out a 5 to get:

5 * 231

The sum of the digits in 231 is 6, which is a number divisible by 3, so 3 must be a factor of 231. Breaking out a 3, we get:

5 * 3 * 77

Lastly, 77 = 7 * 11, so the prime factorization of 1155 is:

1155 = 3 * 5 * 7 * 11

Now we use the method from the Counting Factors of Large Numbers lesson to determine the total number of positive factors in 1155. Since there are no exponents written next to each prime factor, the exponent of each factor must be 1. We then add 1 to each number, giving 2, and then do:

2 * 2 * 2 * 2 = 16

So there are 16 total factors in 1155, which corresponds to 16 unique values for (P - Q) and (P + Q). Since (P − Q) and (P + Q) have 16 unique values, P and Q also must have 16 unique positive values. However, since (P − Q) and (P + Q) can both be negative as well, that accounts for another 16 unique values of P and Q. Because there are 16 positive and 16 unique value pairs for P and Q, the total number of unique coordinate points (P,Q) is 32.

KarishmaB @mickemcgarry MartyMurray, any expert. I do not understand what this question is even asking for. I saw the link to MGMAT problem Bunuel shared, but I still the basic question, what is this question even asking for?

Hi.

What aspect of the question is confusing you?
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Hi Marty, thanks for your response. Let me clarify. What is the relation of coordinate with this question?
Sure, I too got until this point (p+q)(p-q) = 1155 = 11 x 7 x 3 x 5
Then I started to figure out how to fit in the factors in (p+q) and (p-q) format. Then I got stuck and started looking for the solution.
My basic question is that if this question is asking about the total number of unique factors of 1155. How can that be inferred from this question?

Then, if yes, how does it satisfy the values of both p and q?

Also, in case of unique factors, it would be (1+1) x (1+1) x (1+1) x (1+1) = 16. Why is it 32?

Asking how many coordinate points work is an indirect way of asking how many pairs of factors work.

The question didn't need to use coordinate points. It's just a typical Manhattan Prep practice question with multiple layers. So, we need to first figure out how many possible pairs there are and then determine how many points on the coordinate plane work.


Well, P and Q themselves are not the factors. They are values that add up and subtract to the factors.


Because points with two negative coordinates work as well.
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Hi Marty, thanks for your response. Let me clarify. What is the relation of coordinate with this question?
Sure, I too got until this point (p+q)(p-q) = 1155 = 11 x 7 x 3 x 5
Then I started to figure out how to fit in the factors in (p+q) and (p-q) format. Then I got stuck and started looking for the solution.
My basic question is that if this question is asking about the total number of unique factors of 1155. How can that be inferred from this question?[/quote]
Asking how many coordinate points work is an indirect way of asking how many pairs of factors work.

The question didn't need to use coordinate points. It's just a typical Manhattan Prep practice question with multiple layers. So, we need to first figure out how many possible pairs there are and then determine how many points on the coordinate plane work.


Well, P and Q themselves are not the factors. They are values that add up and subtract to the factors.


Because points with two negative coordinates work as well.[/quote]

Got it, thanks. So there’s no need to find out wharf value of p+q and p-q is valid with the factors. For example for 1155 = 35 x 33, I could not find anything satisfying values of integers p and q.

Posted from my mobile device

What about 34 and 1?

Posted from my mobile device
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Understood, missed it.

Check this post first: https://anaprep.com/number-properties-a ... m-product/

Keep in mind:
When can we write a number as difference of squares?
– When the number is odd
or
– When the number has 4 as a factor

We need those integral co-ordinate points (P, Q) for which P^2 - Q^2 = 1155.
Since 1155 is an odd integer, we can write it as the difference of squares. But in how many different ways?


1155 = (P+Q)(P-Q)
In how many different ways can we write 1155 as product of two factors? We can write it as 1155 * 1 or 77 * 15 etc. How will we get all such pairs? By prime factorising 1155.

1155 = 3*5*7*11
So it has 2*2*2*2 = 16 factors which give us 8 pairs such as 1155 * 1, 385 * 3, 231 * 5 etc. Each such pair will give us one distinct value for (P, Q). For example when P+Q is 1155 and P - Q = 1, P = 578 and Q is 577 etc.

So we have found 8 pairs of values for (P, Q) in which P and Q are positive integers. But our question requires them to be integers, not necessarily positive. Since they are squared, any negative sign that either P or Q has or both have, will go away.

Hence, each pair gives us 4 co-ordinate points say (P, Q) can be (578, 577) or (-578, 577) or (578, -577) or (-578, -577)
Since we have 8 such pairs, we get distinct 32 co-ordinate points.

Answer (C)
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Since 1155 is an odd integer, all of its factors must be odd integers.

The difference between any two odd integers is an even integer, and the midpoint between any two odd integers is an integer that's equidistant from the two odd integers.

So, for any two odd integers, there are two integers P and Q such that P + Q and P - Q are the two odd integers, with P being the midpoint between the two odd integers and Q being the distance from the midpoint to each of the integers.

You can try any two odd integers to see this work. For example

13, 41

The difference is 28, the midpoint is 27, and the distance between 27 and each the odd integers is 14. So, for 13 and 41, P and Q are 27 and 14.

So, for any two factors of 1155, there are two integers, P and Q, such that P + Q is one of the factors and P - Q is the other.
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