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If a positive integer n, divided by 5 has a remainder 2

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If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post Updated on: 06 Jul 2017, 08:20
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If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2

A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III

Originally posted by chiccufrazer1 on 20 Mar 2013, 15:14.
Last edited by Bunuel on 06 Jul 2017, 08:20, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 20 Mar 2013, 15:40
2
If a positive integer n,divided by 5 has a remainder 2,which of the following must be true
I. n is odd
II. n+1 cannot be a prime number
III. (n+2)divided by 7 has remainder 2

Some valid values for n: 7, 12, 17, 22, 27, 32... or, in other words: \(n=(i * 5) + 2\) for i=1,2,3...

I. FALSE: we see that n can we odd or even.
II. FALSE: (n+1) could be a prime number. Example: n=12 --> (n+1)=13 is prime. Other example: for n=22, (n+1)=23 is prime.
III. FALSE: for n=12, (n+2)=14, divided by 7 has remainder zero.

None answer is true.

Answer: A
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 20 Mar 2013, 20:53
I. 22 and 27 both have a reminder of 2. So False (Options remaining - A,D,E)
II. 22 + 1 = 23 is a prime number. So False (Options remaining - A)

Ans (A)
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 20 Mar 2013, 20:54
chiccufrazer1 wrote:
If a positive integer n,divided by 5 has a remainder 2,which of the following must be true I. n is odd
II. n+1 cannot be a prime number
III. (n+2)divided by 7 has remainder 2

A.none
B.I only
C.I and II only
D.II and III only
E.I,II and III


n can be written as :

n = 5k+2. Thus, taking k=0, we have n=2.

I.n=2,even.False
II.2+1=3, is a prime. False.
III.n+2 = 4,4 divided by 7 leaves a remainder of 4. False.

A.
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 21 Mar 2013, 00:36
chiccufrazer1, you forgot to provide the OA in your post. Just make sure you do provide it for your future problems. :)

Alright, let's solve this :

We know that n, a positive integer, yields a remainder of 2 when divided by 5. So according to the algebraic form of the division operation, we'll have :

\(n = 5*q + 2\) with q being a positive integer as well.

This expression allows us to give out some valid possibilities for n by playing with the value of q, such as :

q = 0 => n = 2
q = 1 => n = 7
q= 2 => n =12

Now, from these first values we can already cross off statement I.(n is odd) , since n can be 7 (which is odd) or n can be 12 (which is even).

Statement II. (n+1 cannot be a prime number) can also be crossed off. Consider n = 12, which is not a prime number and yields a remainder of 2 when divided by 5. If we add 1 to it, we get 13, which IS a prime number, so that contradicts statement II.

Finally, statement III. (n+2 yields a remainder of 2 when divided by 7) can also be crossed off. Again consider n = 12. Add 2 to it and we get a 14 which is a multiple of 7.

In short, all statements have been contradicted and the correct answer choice to the question is A : none of the statements above are true.

Hope that helped. :-D
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 21 Mar 2013, 03:16
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2


A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III[/quote]

A positive integer n, divided by 5 has a remainder 2 --> \(n=5q+2\), so n could be 2, 7, 12, 17, 22, 27, ...

I. n is odd. Not necessarily true, since n could be 2, so even.

II. n+1 cannot be a prime number. Not necessarily true, since n could be 2, so n+1=3=prime.

III. (n+2) divided by 7 has remainder 2. Not necessarily true, since n could be 2, so n+2=4 and 4 divided by 7 has remainder 4.

Answer: A.

Hope it's clear.
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If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post Updated on: 06 Jul 2017, 08:24
1
Bunuel wrote:
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2


A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III


Quote:
A positive integer n, divided by 5 has a remainder 2 --> \(n=5q+2\), so n could be 2, 7, 12, 17, 22, 27, ...

I. n is odd. Not necessarily true, since n could be 2, so even.

II. n+1 cannot be a prime number. Not necessarily true, since n could be 2, so n+1=3=prime.

III. (n+2) divided by 7 has remainder 2. Not necessarily true, since n could be 7, so n+2=9.

Answer: A.

Hope it's clear.


Bunuel ,
I am confused by your analysis of III: (n+2) divided by 7 has remainder 2

If n = 7 and (n + 2) = 9, then \(\frac{9}{7}\) = 1 + R2.

n could be 2, 7, 12, 17 ...

If n = 12, then (n+2) = 14, which, when divided by 7, leaves remainder 0.

If n = 17, (n+2) = 19, which, when divided by 7, leaves remainder 5.

Those two examples (or others) seem to me to be what should be used to show that III does not satisfy the condition "must be true."

The one you chose proves that III could be true; I'm having a hard time understanding how n = 7 proves that III does not have to be true. Am I missing something?

Originally posted by generis on 06 Jul 2017, 08:11.
Last edited by generis on 06 Jul 2017, 08:24, edited 1 time in total.
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 06 Jul 2017, 08:21
genxer123 wrote:
Bunuel wrote:
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2


A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III


Quote:
A positive integer n, divided by 5 has a remainder 2 --> \(n=5q+2\), so n could be 2, 7, 12, 17, 22, 27, ...

I. n is odd. Not necessarily true, since n could be 2, so even.

II. n+1 cannot be a prime number. Not necessarily true, since n could be 2, so n+1=3=prime.

III. (n+2) divided by 7 has remainder 2. Not necessarily true, since n could be 7, so n+2=9.

Answer: A.

Hope it's clear.


Bunuel ,
I am confused by your analysis of III: (n+2) divided by 7 has remainder 2

If n = 9, then \(\frac{9}{7}\) = 1 + R2.

n could be 2, 7, 12, 17 ...

If n = 12, then (n+2) = 14, which, when divided by 7, leaves remainder 0.

If n = 17, (n+2) = 19, which, when divided by 7, leaves remainder 5.

Those two examples (or others) seem to me to be what should be used to show that III does not satisfy the condition "must be true."

The one you chose proves that III could be true; I'm having a hard time understanding how n = 7 proves that III does not have to be true. Am I missing something?


You are right. Edited the question.
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 06 Jul 2017, 10:10
chiccufrazer1 wrote:
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2

A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III


Possible values of n are { 7 , 12 , 17 , 22 , 27 ................... }

Now, check the options -

I. n can be Odd/Even
II. n can be Prime / Non Prime
III. n can/can not have remainder 2

Thus, the answer will be (A)
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 14 Jul 2017, 09:40
chiccufrazer1 wrote:
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2

A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III


We can express n as:

n = 5q + 2

Let’s now analyze each Roman numeral:

I. n is odd

If q = 2, then 5q + 2 = 12, so n does not have to be odd.

II. n+1 cannot be a prime number

If q = 2, then 5q + 2 = 12, so n + 1 = 13, which is prime. So II does not have to be true.

III. (n+2) divided by 7 has remainder 2

n + 2 = 5q + 4

If q = 2, then 5q + 4 = 14, which has a remainder of zero when divided by 7. So III does not have to be true.

Answer: A
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Re: If a positive integer n, divided by 5 has a remainder 2  [#permalink]

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New post 13 Nov 2017, 14:12
1
Top Contributor
chiccufrazer1 wrote:
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2

A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III


-----------------ASIDE----------------------------------
When it comes to remainders, we have a nice rule that says:

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
------ONTO THE QUESTION!!!------------------------

Positive integer n, divided by 5 has a remainder 2
Some possible values of n: 2, 7, 12, 17, 22, 27, 32, 37, . . . etc

Now let's examine the statements:
I. n is odd.
This need not be true.
Among the possible values of n, we see that n need not be odd
So statement 1 is FALSE

II. n+1 cannot be a prime number.
Not true.
Among the possible values of n, we see that n COULD equal 2
2+1 = 3, and 3 IS a prime number
So, n+1 CAN BE a prime number
So statement 2 is FALSE

NOTE: Since statements I and II are false, we need not examine statement III, since there are no answer choices that suggest that only statement III is true.
So, the correct must be A

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Re: If a positive integer n, divided by 5 has a remainder 2 &nbs [#permalink] 13 Nov 2017, 14:12
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