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

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Re: If a positive integer n, divided by 5 has a remainder 2, which of the [#permalink]
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A positive integer n, divided by 5 has a remainder 2
n = 5x +2..........eqn (1)

So, n can be 2, 7, 12 ,17, 22, 27, 32 and so on

Statement I : n is odd
Substituting x= 2 in eqn (1)
n =12
So, n doesn't need to be odd . Not always true

Statement II : n + 1 cannot be a prime number
If we take n = 2, n +1 =3 which is a prime number
Not true

Statement III : (n + 2) divided by 7 has remainder 2
If we take n = 12 , remainder of (n + 2) divided by 7 is 0
Not true

Correct Option: A. none
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Re: If a positive integer n, divided by 5 has a remainder 2 [#permalink]
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]
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.

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

RELATED VIDEO

Originally posted by BrentGMATPrepNow on 13 Nov 2017, 15:12.
Last edited by BrentGMATPrepNow on 17 May 2021, 08:36, edited 1 time in total.
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If a positive integer n, divided by 5 has a remainder 2 [#permalink]
Top Contributor
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

When it comes to remainders, we have a nice property 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.

Given: When positive integer n is divided by 5, we get remainder 2
So, the possible values of n are: 2, 7, 12, 17, 22, 27, 32, . . . .

Now let's examine each statement....
I. n is odd
If n = 2, then n is NOT odd.
So, statement I need not be true.
Check the answer choices.... eliminate B, C and E, since they state that statement I is true.

II. n + 1 cannot be a prime number
If n = 2, then n+1 = 3, and 3 is prime.
Check the remaining answer choices.... eliminate D, since it states that statement I Iis true.

By the process of elimination, the correct answer is A.

ASIDE: Notice that we are able to arrive at the correct answer without having to analyze statement III.
We were able to do this because we eliminated incorrect answer choices after analyzing each statement.

Cheers,
Brent

RELATED VIDEO

Originally posted by BrentGMATPrepNow on 12 Mar 2020, 07:33.
Last edited by BrentGMATPrepNow on 17 May 2021, 09:41, edited 1 time in total.
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Re: If a positive integer n, divided by 5 has a remainder 2, which of the [#permalink]
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

N can take on values such as 2, 7, 12, 17, 22, etc.

Looking at the answer choices, we see that none of I, II, or III must be true.

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

Theory: Dividend = Divisor*Quotient + Remainder

n -> Dividend
5 -> Divisor
a -> Quotient (Assume)
2 -> Remainder

=> n = 5*a + 2 = 5a + 2

Now, lets take option choice and see if we can prove it wrong

I n is odd
n = 5a + 2
If we take a = 0. We get the value of n as
n = 0*2 + 2 = 0 + 2 = 2 => NOT odd => FALSE

II n + 1 cannot be a prime number
Same example as first one
We can have n = 2 => PRIME => FALSE

III (n + 2) divided by 7 has remainder 2
n = 5a + 2
=> n + 2 = 5a + 2 + 2 = 5a + 4
a = 0
n + 2= 5*0 + 4 = 4 => Remainder of n (=4) by 7 becomes 4 => FALSE