Answer: Option C

Please check the video for the step-by-step solution.

GMATinsight's Solution





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We can have multiple possibilities here clearly. (p=10, q=15) or (p=20,q=25) and so on. So, (1) alone is not sufficient


Just like with the previous statement, we can have multiple possibilities here (10,17) or (20,27) and so on.

Now, let's consider both the statements together...
Since there is a remainder in either case, we can conclude p and q are not equal. So, either p>q or q>p. Let's consider these cases one after another

Case 1: if p>q
In this case, the remainder of q/p will be equal to q. A number divided by something greater than itself, leaves the entire dividend as remainder. From statement 2, we know that quantity is going to be 7. In other words q = 7.
Now, we have to check if this is even a possibility that satisfies statement (1). q=7 and p=12 satisfies (1). So, yes, this is a possibility

Case 2: if q>p
In this case, the remainder of p/q will be equal to p. From (1) we get p = 5
Now, just like in the previous case, let's see if there's even a possibility that this satisfies the statement (2)
(2) says that any number (q) divided by p(=5) leaves a remainder of 7. Clearly, this is impossible. Becaue 5 can divide 7 at least one more time. In other words, the remainder can never be greater than the divisor. So, case 2 is not possible.

Thus, we have only 1 possibility of q (from case 1) and this is q = 7.
So, (1) & (2) together sufficient to give the value of q. Answer : C

What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.
St 1: p=qK + 5, where k = constant p can be 5 and q can be any value>5 so no sufficient

(2) When q is divided by positive integer p, the remainder is 7.
St 1: q=pK + 7, where k = constant q can be 7 and 7 can be any value>7
also what if q = 15 and p = 8 so two value not sufficient

After combining both st.
we get to know that possible value for p = 12 and q = 7 so both sufficient

IMO C

St 1
p= Mq+ 5
q can be anything greater than 5
Insufficient

St.2
q = Np+ 7
q can be anything
Insufficient

Combining St1 and St2
q = N(Mq+5)+7
q = NMq + N5 + 7
q = (5N + 7) / (1-NM)
q is +ve
Hence NM = 0. So N or M or both can be zero
Not sufficient
Hence E

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here is my solution. Not sure if this is smart
Let x,y be some integer quotients
\(p=qx+5\).............1
\(q=py+7\).............2
but substituting eq 1 in 2
we get
\(q=\frac{(5y+7)}{(1-xy)}\)
it is not sufficient even if bothe options are taken into consideration
so E

(1) When positive integer p is divided by q, the remainder is 5.
-- Considering statement 1 alone
=> p = qx + 5
from this we know q is greater than 5. But don't know the exact value.
-- Not sufficient.

(2) When q is divided by positive integer p, the remainder is 7.
=> q = py + 7
p is greater than 7.
-- Not sufficient.

On combining both the statement,
Case 1 : P =5, then remained for q will not be 7, it should become 2.
Case 2: q = 7, then P can take any value for which we will get remainder 5.

On combining we will get answer. Therefore, C is the answer.

What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.- insufficient because we can get multiple values of p and q. for e.g. q=6, p=11; q=7, p=12 etc.

(2) When q is divided by positive integer p, the remainder is 7.- insufficient because we can get multiple values of p and q. for e.g q=15, p=8; q=16,p=9 etc.

together taking (1) & (2) also we get multiple values of p and q. so the answer is e.

Asked: What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.
pmodq = 5
p = qk + 5; where k Is an integer.
q > 5;
Since p is unknown, the value of q can not be found
NOT SUFFICIENT

(2) When q is divided by positive integer p, the remainder is 7.
qmodp = 7
q = pm + 7; where m Is an integer.
p > 7;
q can not be determined.
NOT SUFFICIENT

(1) + (2)
(1) When positive integer p is divided by q, the remainder is 5.
(2) When q is divided by positive integer p, the remainder is 7.
q = (qk + 5)m + 7 = qkm + 5m + 7
q(1-km) = 5m + 7
q = (5m+7)/(1-km) > 5
Since 5m + 7 > 0 ; 1-km > 0 ; km < 1
It is only possible when km=0
If k=0; p=5; not possible since p>7
If m=0; q = 7; p = 7k + 5; possible
q = 7
SUFFICIENT

IMO C
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(1) When positive integer p is divided by q, the remainder is 5.

So p=qx+5
Insufficient
(2) When q is divided by positive integer p, the remainder is 7.
Q=py+7 (y is number of times divided)
Still insufficient
IMO E

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Stat 1- p/q = 5, not sufficient to get value of q.

Stat 2- q/p= 7, not sufficient to get value of q, but p>7

Combining both, from stat 2, p= 8, 9, 11, 12, 13...then making stat 1 holds true, q= 3, 4, 6, 7, 8...
Now, to hold stat 2 true as well (p,q)= (12,7). Sufficient to get q value.

So, I think ans. C.

Give in question q > 0
Statement 1 in not sufficient as p and q can be any value but for statement
P = qx + 5
1 we got Q> 5

2 statement is not sufficient as P and q can have multiple values
q = pY + 7 so P> 7
Combining both still is not sufficient to answer
E is correct

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(1) When positive integer p is divided by q, the remainder is 5.

p and q can be any number. Hence insufficient.

Example,
P/Q - 5/7
P/q - 5/13

Hence insufficient.

(2) When q is divided by positive integer p, the remainder is 7.

Q/p - 7/12
Q/p - 20/13
Hence insufficient.

Combining 1 & 2,

P/q - 12/7 - reminder 5
Q/p - 7/12 - reminder 7

Hence C

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what is the value of q?? q is a +ve I

A) P/q ---remainder is 5, now you know that remainder is always less than divisor, so q>5. q can attain value - insufficient

b) q/p ---remainder is 7, now by same logic, p>7. what values can q take? more than one. if p=8, q can be 7, 15, 22 and so on.....insufficient

now lets take a and b together shall we?

from a) we can write p = xq +5 where x is the quotient. we know that p >7, so x cannot be zero. now x can be greater than or equal to 1. In that case, it means that p is always greater than q. Now its getting interested, isnt it?

come to the 2nd situation. p>q, what can we conclude?........that q=7.....clear? No....lets take several examples, shall we?

6/8 - what will be the remainder?

7/9- what will be the remainder?

now p>q and we know the remainder of q/p. Over, we are done and dusted. I wont reveal the answer. Its straight-forward. Let there be some excitement. Well the answer is in this explanation only.

What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.

This means q>5, INSUFFICIENT

(2) When q is divided by positive integer p, the remainder is 7.
p>7, INSUFFICIENT

1 + 2,

Either of the two values have to be greater and gives integer,
and in case we take reciprocal I don’t see how we should get integer again.
IMO – E.

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What is the value of positive integer q?

Answer: C

(1) When positive integer p is divided by q, the remainder is 5.
We know, by remainder theorem
the statement can be written as
p=xq+5
where x is any integer value(quotient) starting from 0 and positive integers

But we cannot assume any 1 particular value of y from this
but we know
q>5

Statement 1 is not sufficient


(2) When q is divided by positive integer p, the remainder is 7.
We know, by remainder theorem
the statement can be written as
q=yp+7

where y is any integer value(quotient)

But we cannot assume any 1 particular value of y from this
but we know
from here that q>=7

Statement 2 is insufficient

Statement 1 and 2 combined

We have 2 eqns
p=xq+5 - eqn 1
q=yp+7 - eqn 2

First putting x=0
we will get
p=5 but this value will not satisfy eqn 2
as then remainder should be 5
So P>5
and it means P>7 also from Eqn 2
hence P has many values but all are greater than 7
So from this we get that Q=7
and P can have many values like=12,19,26.....................
So we have only one value of Q i.e. 7

Both are sufficient

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Find value of q.

1.p/q, Remainder = 5.
So q is greater than 5.

2.q/p, Remainder = 7.
p > 7

Answer E. As there are many combination of q that are greater then 5 and combination of p greater than 7
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What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.
(2) When q is divided by positive integer p, the remainder is 7.

Statement 1:

When p=12, q=7 then r=5
When p=18, q=13 then r=5
No unique value of q. So statement 1 alone is not sufficient to answer the question.

Statement 2:

When p=13, q=7 then r=7
When p=6, q=13 then r=7
No unique value of q. So statement 2 alone is not sufficient to answer the question.

Combining both statement 1 & statement 2, we can infer that q cannot have unique value.
So option E is correct answer.

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What is the value of positive integer q?

(1) When positive integer p is divided by q, the remainder is 5.

(2) When q is divided by positive integer p, the remainder is 7.

Take the couple {p, q} = {12, 7} --> (1) and (2) pass --> q = 7
But take another couple {p, q} = {7, 12} --> (1) and (2) pass --> q = 12
--> Both (1) and (2) INSUFFICIENT

So answer should be E

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Answer:E
Statement 1:not sufficient
Given condition p=qk+5
For above condition to be valid q>5 , q must be greater than 5.
Hence q=6, 7,8,9.....,15,16,.....
Not sufficient not a unique value
Statement 2:not sufficient
Satisfying given condition q=pj+7
For above condition p>7, means p=8, 9,10....
Now when p=8, q may be q=7, 15,23,.....
Again q doesn't have unique value. Not sufficient
Combining 1+2, not sufficient
From both statement common value of q=7, 15....
Hence not unique integer

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