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Set P consists of the first n positive multiples of 3 and set Q

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Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 23 Mar 2019, 09:00
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  95% (hard)

Question Stats:

17% (04:06) correct 83% (02:41) wrong based on 24 sessions

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Set P consists of the first n positive multiples of 3 and set Q consists of the first m positive multiples of 5. The sum of all the numbers in set P is equal to R and the sum of all the numbers in set Q is equal to S. If 'm 'and 'n' are positive integers, is the difference between R and S odd?

I. m is odd and n is even
II. m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer
Spoiler: OA
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Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post Updated on: 24 Mar 2019, 09:54
kiran120680 wrote:
Set P consists of the first n positive multiples of 3 and set Q consists of the first m positive multiples of 5. The sum of all the numbers in set P is equal to R and the sum of all the numbers in set Q is equal to S. If 'm 'and 'n' are positive integers, is the difference between R and S odd?

I. m is odd and n is even
II. m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer


As it can be hard to understand what we're dealing with just by reading, we'll first break things down into bite-size chunks.
This is a Precise approach.

We know that
R = 3 + 6 + 9 + ... for n multiples so R = 3(1 + 2 + 3... n), i.e. R is 3 * the sum of the first n numbers.
Similarly, S = 5 * the sum of the first m numbers.
Since R - S is odd only if both R,S are even or both are odd, we need to know if the sums 1+2+...n and 1+2+...m are even or odd.
Using the standard arithmetic sum, 1+2+...n = n(n+1)/2 which is even if n or n+1 is a multiple of 4 and odd otherwise.
Now that we know exactly what to look for, let's look at our statements.

(1) does not tell us if (m or m+1) and (n or n+1) are multiples of 4.
(2) tells us that m+1 is a multiple of 4 but does not tell us if n or n+1 are a multiple of 4.
Combining gives us no information on whether n or n+1 is a multiple of 4.

(E) is our answer.
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Originally posted by DavidTutorexamPAL on 23 Mar 2019, 12:41.
Last edited by DavidTutorexamPAL on 24 Mar 2019, 09:54, edited 1 time in total.
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Re: Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 24 Mar 2019, 09:40
DavidTutorexamPAL wrote:
kiran120680 wrote:
Set P consists of the first n positive multiples of 3 and set Q consists of the first m positive multiples of 5. The sum of all the numbers in set P is equal to R and the sum of all the numbers in set Q is equal to S. If 'm 'and 'n' are positive integers, is the difference between R and S odd?

I. m is odd and n is even
II. m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer


As it can be hard to understand what we're dealing with just by reading, we'll first break things down into bite-size chunks.
This is a Precise approach.

We know that
P = 3 + 6 + 9 + ... for n multiples so P = 3(1 + 2 + 3... n), i.e. P is 3 * the sum of the first n numbers.
Similarly, Q = 5 * the sum of the first m numbers.
Since R - S is odd only if both R,S are even or both are odd, we need to know if the sums 1+2+...n and 1+2+...m are even or odd.
Using the standard arithmetic sum, 1+2+...n = n(n+1)/2 which is even if n or n+1 is a multiple of 4 and odd otherwise.
Now that we know exactly what to look for, let's look at our statements.

(1) does not tell us if (m or m+1) and (n or n+1) are multiples of 4.
(2) tells us that m+1 is a multiple of 4 but does not tell us if n or n+1 are a multiple of 4.
Combining gives us no information on whether n or n+1 is a multiple of 4.

(E) is our answer.


Little confused, is there any better way to solve this question.
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Re: Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 24 Mar 2019, 09:52
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mangamma wrote:

Little confused, is there any better way to solve this question.


Hey mangamma,

The logic of this question is indeed hard.
Try using numbers to make things concrete; this is a useful tool for approaching questions you don't really get.
In this case:

(1) trying m = 1 and n = 2 gives R = 3+6 = 9 and Q = 5. Then R - S = 9 - 5 which is even. trying m = 3 and n = 2 gives R = 9 and S = 5 + 10 + 15 = 30. Then R - S = 30- 9 = 21 which is odd.
Insufficient.
(2) so we can still pick m = 3 and n = 2, giving R-S odd. We'll keep m = 3 and change to n = 4, giving S =30 like before and R = 3+6+9+12 = 30. Then R-S = 30 - 30 = 0 which is even.
Insufficient.

Combined:
Both examples given in (2), i.e. (m,n)=(3,2) and (m,n)=(3,4) work when combining the statements as well.
So still insufficient.

Hence (E).

Also, I had a typo in the original explanation which I have now fixed. Hope it is clearer :)
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Re: Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 24 Mar 2019, 09:58
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DavidTutorexamPAL wrote:
mangamma wrote:

Little confused, is there any better way to solve this question.


Hey mangamma,

The logic of this question is indeed hard.
Try using numbers to make things concrete; this is a useful tool for approaching questions you don't really get.
In this case:

(1) trying m = 1 and n = 2 gives R = 3+6 = 9 and Q = 5. Then R - S = 9 - 5 which is even. trying m = 3 and n = 2 gives R = 9 and S = 5 + 10 + 15 = 30. Then R - S = 30- 9 = 21 which is odd.
Insufficient.
(2) so we can still pick m = 3 and n = 2, giving R-S odd. We'll keep m = 3 and change to n = 4, giving S =30 like before and R = 3+6+9+12 = 30. Then R-S = 30 - 30 = 0 which is even.
Insufficient.

Combined:
Both examples given in (2), i.e. (m,n)=(3,2) and (m,n)=(3,4) work when combining the statements as well.
So still insufficient.

Hence (E).

Also, I had a typo in the original explanation which I have now fixed. Hope it is clearer :)


Now its claer. Thanks for the explanation.
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Re: Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 24 Mar 2019, 10:14
DavidTutorexamPAL wrote:
mangamma wrote:

Little confused, is there any better way to solve this question.


Hey mangamma,

The logic of this question is indeed hard.
Try using numbers to make things concrete; this is a useful tool for approaching questions you don't really get.
In this case:

(1) trying m = 1 and n = 2 gives R = 3+6 = 9 and Q = 5. Then R - S = 9 - 5 which is even. trying m = 3 and n = 2 gives R = 9 and S = 5 + 10 + 15 = 30. Then R - S = 30- 9 = 21 which is odd.
Insufficient.
(2) so we can still pick m = 3 and n = 2, giving R-S odd. We'll keep m = 3 and change to n = 4, giving S =30 like before and R = 3+6+9+12 = 30. Then R-S = 30 - 30 = 0 which is even.
Insufficient.

Combined:
Both examples given in (2), i.e. (m,n)=(3,2) and (m,n)=(3,4) work when combining the statements as well.
So still insufficient.

Hence (E).

Also, I had a typo in the original explanation which I have now fixed. Hope it is clearer :)


I too little confused with the solution. Now I understood. Thank you David :thumbup:
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Set P consists of the first n positive multiples of 3 and set Q  [#permalink]

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New post 24 Mar 2019, 11:17
kiran120680 wrote:
Set P consists of the first n positive multiples of 3 and set Q consists of the first m positive multiples of 5. The sum of all the numbers in set P is equal to R and the sum of all the numbers in set Q is equal to S. If 'm 'and 'n' are positive integers, is the difference between R and S odd?

I. m is odd and n is even
II. m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer


Here is how I solved. This is a Yes/No question and you need to find all the areas to see if you are getting consistent answers.

By looking at the statements, I got an idea that while testing conditions, n has to be odd and m has to be even! So I took two examples so it'll be easy to test/solve:
Example 1: P={3,6,9} => n=3 and ΣP=18=R & Q= {5,10} =>m=2 and ΣQ=15=S
Nowhere in the questions it is mentioned that the multiples should be consecutive. So I'm gonna take different numbers now.
Example 2: P={3,9,15} => n=3 and ΣP=27=R & Q= {10,15} =>m=2 and ΣQ=25=S

Another hint: We gotta find if R-S=odd. As per number property rules, odd-even=odd or even-odd=odd(keep this in kind while taking examples. It'll help!)

Moving onto the statements

Statement (1): m is odd and n is even
Now the role of examples come in. From example 1, R-S= 18-15=3 ODD......YES
example 2, R-S= 27-25=2 EVEN...........NO
Different answers.....INSUFFICIENT

Statement (2): m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x
m=4x+3=odd (odd+even=odd) and n=2x= even
Now this statement is as same as statement 1
Different answers.....INSUFFICIENT

Combining statement (1) + (2) would be same as above. Different answers. INSUFFICIENT

Hence, answer is option E
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Set P consists of the first n positive multiples of 3 and set Q   [#permalink] 24 Mar 2019, 11:17
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