If n is a positive integer, then is (n + 1)(n + 3) a multiple of 4?

(1) (n + 2)(n + 4) is odd.

If (n + 2)(n + 4) is odd, then the value of (n + 2) is odd and the value of (n + 4) is odd. If integers (n + 4) and (n + 2) are odd then n is odd. Since n is odd, let it be of the form 2k+1, where k is an non negative integer.

Then (n + 1)(n + 3) = (2k+1+1)*(2k+1+3) = (2k+2)*(2k+4) = 2(k+1)*2(k+2) = 4*(k+1)*(k+2) = 4*Integer.
So, (n + 1)(n + 3) is definitely a multiple of 4.

Statement 1 is sufficient to answer the question.

(2) (n + 3)(n + 6) is even.

If (n + 3)(n + 6) is even
Case 1: The value of (n + 3) is odd and the value of (n + 6) is even. So, the value of n is even. If the value of n is even, the value of (n + 1)(n + 3) is odd and not a multiple of 4(An odd number can never be a multiple of 4 or even 2).
Case 2: The value of (n + 3) is even and the value of (n + 6) is odd. So, the value of n is odd. If the value of n is odd, the value of (n + 1)(n + 3) = (2k+1+1)*(2k+1+3) = (2k+2)*(2k+4) = 2(k+1)*2(k+2) = 4*(k+1)*(k+2) = 4*Integer.

Two different answers. So, statement 2 is insufficient to answer the question.

IMO A

Is (50 + 5n)/n^2 an integer?

Assumption in RED.

(1) n/5 is a positive integer.

So, n = 5k where k is a positive integer
(50 + 5n)/n^2
= (50 + 5*5k)/(5k)^2
= (25*(2+k))/(25*(k^2))
= (2+k)/(k^2)
If k = 1 then the value of (2+k)/(k^2) = 3, an integer.
If k = 4 then the value of (2+k)/(k^2) = 3/8, not an integer.
Two different answers. So, statement 1 is insufficient to answer the question.

(2) n/10 is a positive integer

So, n = 10k where k is a positive integer
(50 + 5n)/n^2
= (50 + 5*10k)/(10k)^2
= 50(1+k)/(100*(k^2))
= (1+k)/(2*k*k)
If k = 1 then the value of (1+k)/(2*k*k) = 1, an integer.
If k = 4 then the value of (1+k)/(2*k*k) = 5/32, not an integer.
Two different answers. So, statement 2 is insufficient to answer the question.

From 1 and 2

If n = 10, then (50 + 5n)/n^2 = (50+50)/100 = 1, an integer.
If n = 40, then (50 + 5n)/n^2 = (50+200)/1600 = 5/32, not an integer.
Two different answers. So, statement 1+2 combined is insufficient to answer the question.
IMO E

What is p/q? ( I'm not sure but I think there is a mistake in statement 2)

(1) (p + q/p - q) + (p - q/p + q)= 5 1/5

(2) (p - q/p + q) - (p - q/p + q)= 4 4/5

Is the term (p + q/p - q) + (p - q/p + q) same as (p + (q/p) - q) + (p - (q/p) + q) or is it ((p + q)/(p - q)) + ((p - q)/(p + q)) ?

Thanks for the wonderfull explanations.

Regarding the 1st question, it's ((p + q)/(p - q)) + ((p - q)/(p + q))
soory for that

No worries mate! happens!

What is p/q?

(1) ((p + q)/(p - q)) + ((p - q)/(p + q))= 5 1/5

((p + q)/(p - q)) + ((p - q)/(p + q))= 26/5
((p + q)^2 + (p - q)^2)/(p^2 - q^2)= 26/5
((p^2 + q^2 + 2pq)+ (p^2 + q^2 - 2pq))/(p^2 - q^2)= 26/5
2*(p^2 + q^2)/(p^2 - q^2) = 26/5
10*(p^2 + q^2)=26*(p^2 - q^2)
10p^2 + 10q^2 = 26p^2 - 26q^2
36q^2 = 16 p^2
p^2/q^2 = 36/16 = (6/4)^2
p/q = 6/4 or -6/4
p/q = 3/2 or -3/2.
So the value of p/1 can be 3/2 or -3/2. We have two different answers. So, Statement 1 is sufficient to answer the question.

(2) ((p - q)/(p + q) - (p - q)/(p + q))= 4 4/5

((p - q)/(p + q) - (p - q)/(p + q))= 4 4/5
0 = 4 4/5?
As you doubted, there is a mistake in statement 2. Can you recheck and revert ?

First question is wrong

Second question

If n is od the product of (n+1)*(n+3) is definitely a multiple of 4

S2 is not sufficient

(A) is answer

Third question: none of the given conditions yield any info about 50/n^2 + 5/n
(E)

alnasser973 wrote:These are some problems that I don't digest their answers explanations very well, please if any one could help me I will be so thankful )

If n is a positive integer, then is (n + 1)(n + 3) a multiple of 4?

(1) (n + 2)(n + 4) is odd.

(2) (n + 3)(n + 6) is even.

Lets look closely at (n + 1)(n + 3). This is a product of two integers separated by 2, which means that (n + 1) & (n + 3) are either both ODD CONSECUTIVE integers or EVEN CONSECUTIVE integers.

Statement(1):
If (n + 2)*(n + 4) is ODD, then each of (n + 2) and (n + 4) must be ODD and each of (n + 1) and (n + 3) is EVEN. Thus (n + 1)*(n + 3) is a multiple of 4. ( Do you see that it is a multiple of 8 also?)
SUFFICIENT

Statement(2):
The Statement does not add anything to what we already know. Since (n + 3) and (n + 6) are separated by 3, if one of them is EVEN then the other one is ODD. Their product has to be an EVEN number because EVEN*ODD = ODD*EVEN = EVEN.
The even number between them decides if the product (n + 3)*(n + 6) is divisible by 4 or not. If that even number is a multiple of 4, then their product will also be a multiple of 4 but if that even number is not a multiple of 4, then neither will their product be.
50-50 chances.
INSUFFICIENT

[spoiler](A)[/spoiler] is correct.

alnasser973 wrote: Is (50 + 5n)/n^2 an integer?

(1) n/5 is a positive integer.

(2) n/10 is a positive integer

Statement(1):
n/5 = A,
where A is a positive integer.
Then n = 5A
Substituting 'n' in the question, we get
(50 + 25A)/(25A^2)
= (2 + A)/A^2
This is an integer only when A = 1, otherwise it is not.
INSUFFICIENT

Statement(2):
n/10 = B,
where B is a positive integer.
Then n = 10B
Substituting 'n' in the question, we get
(50 + 50B)/(100B^2)
= (1 + B)/(B^2)
This is an integer only when B = 1, otherwise it is not.
INSUFFICIENT

When the statements are combined, we are looking for 'n' that is a multiple of 5 as well as 10. This means that 'n' is a multiple of 10 and now this is same as what was mentioned in Statement(2). If Statement (2) was insufficient, so will the combination of the two statements be.

[spoiler](E)[/spoiler] is the answer.

Thank you all for replying, that means a lot for me.

Regarding the 1st question, I found the mistake it's on the 2nd statement and here is the modified one,

What is p/q?

(1) ((p + q)/(p - q)) + ((p - q)/(p + q))= 5 1/5

(2) ((p + q)/(p - q)) - ((p - q)/(p + q))= 4 4/5

alnasser973 wrote:Thank you all for replying, that means a lot for me.

Regarding the 1st question, I found the mistake it's on the 2nd statement and here is the modified one,

What is p/q?

(1) ((p + q)/(p - q)) + ((p - q)/(p + q))= 5 1/5

(2) ((p + q)/(p - q)) - ((p - q)/(p + q))= 4 4/5

Now suppose ((p + q)/(p - q)) = x. then ((p - q)/(p + q)) will be 1/x.

So the first equation will become x + 1/x = 26/5
=>(x^2 + 1)/x = 26/5
for this equation x can be +5 or 1/5.
So INSUFFICIENT

Now the second equation will be x - 1/x = 24/5
=>(x^2 - 1)/x = 24/5
for this equation x can be +5 or -1/5.
So INSUFFICIENT

Now combining both the equation we get x = 5.
From this we can obtain p/q ratio which would be 3/2.

so Answer is C

BTW what is the OA?

Thank you brother,

That's how I did it after realizing what the mistake in the Q is.

The OA is C[/spoiler]