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For a positive integer m, [m] is defined to be the remainder when

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For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 09 Jul 2018, 00:56
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[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III

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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 09 Jul 2018, 02:58
MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


I. [3n+1] means remainder when 7*(3n+1) is divided by 3..
21n+7 div by 3. 21n is divisible by 3 and 7 leaves a remainder of 1.....YES
II. [3n]
7*3n is divisible by 3 so 0....not 1.....ans NO
III. [3n]+2
[3n] is divisible by 3, so remainder will be 2....ans NO

Only I

A
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For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 09 Jul 2018, 04:11
MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m]is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


I. [3n+1] is the remainder when 7(3n+1) is divided by 3.
Or, Rem((21n+7)/3)=Rem(21n/3)+Rem(7/3)=0+1=1
So,[3n+1] =1
II.[3n] is the remainder when 7(3n) is divided by 3.
Or,Rem(21n/3)=0
So, [3n]=1
III.From II, we have obtained , [3n]=0
So, [3n]+2=0+2=0

So, the expression in I, yields a value of 1.

Ans. (A)
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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 09 Jul 2018, 12:14
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MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


I. [3n+1] means remainder when 7(3n+1) = 21n + 7 is divided by 3, we get remainder as 1. Hence [3n+1] = 1

II. [3n] means remainder when 7(3n) = 21n is divided by 3, we get remainder as 0. Hence, [3n] = 2

III. [3n] + 2 means remainder when 7(3n) + 2 = 21n + 2 is divided by 3, we get remainder as 2. Hence, [3n]+2 = 2

Hence Only I gives remainder as 1.


Answer A.



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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 11 Jul 2018, 01:10
=>

Statement I
\(7(3n+1) = 21n + 7 = 3(7n+2) + 1\)
Thus, \([3n+1] = 1\)

Statement II
\(7(3n) = 21n = 3*7n + 0\)
Thus, \([3n] = 0\)

Statement III
Since \([3n] = 0\), we have \([3n] + 2 = 2.\)

Thus, only \([3n+1]\) equals \(1\).

Therefore, the answer is A.

Answer: A

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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 14 Jul 2018, 19:22
MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


Let’s analyze each Roman numeral.

I. [3n+1]

[3n+1] is the remainder when 7(3n + 1) = 21n + 7 is divided by 3. Since 21n is divisible by 3, we see that [3n+1] is equal to the remainder when 7 is divided by 3, which is 1. So I is true.

II. [3n]

[3n] is the remainder when 7(3n) = 21n is divided by 3. Since 21n is divisible by 3, we see that the remainder is 0. So II is not true.

III. [3n] + 2

We saw that in Roman numeral II, [3n] has a remainder 0 when it’s divided by 3. So [3n] + 2 has a remainder of 2 when it’s divided by 3. III is not true.

Answer: A
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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 27 Jul 2018, 10:42
MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


MathRevolution is it typical of GMAT type of question ? how often can i encounter such kind of questuons on gmat ?
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Re: For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 27 Jul 2018, 10:48
GyMrAT wrote:
MathRevolution wrote:
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III


I. [3n+1] means remainder when 7(3n+1) = 21n + 7 is divided by 3, we get remainder as 1. Hence [3n+1] = 1

II. [3n] means remainder when 7(3n) = 21n is divided by 3, we get remainder as 0. Hence, [3n] = 2

III. [3n] + 2 means remainder when 7(3n) + 2 = 21n + 2 is divided by 3, we get remainder as 2. Hence, [3n]+2 = 2

Hence Only I gives remainder as 1.


Answer A.



Thanks,
GyM



hey GyMrAT how did you get remainer of 1? would like to see detailed solution of correct answer choice. thanks for taking time to explain :-)
have a great weekend :)

I. [3n+1] means remainder when 7(3n+1) = 21n + 7 is divided by 3, we get remainder as 1. Hence [3n+1] = 1
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For a positive integer m, [m] is defined to be the remainder when  [#permalink]

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New post 27 Jul 2018, 11:21
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dave13 wrote:


hey GyMrAT how did you get remainer of 1? would like to see detailed solution of correct answer choice. thanks for taking time to explain :-)
have a great weekend :)

I. [3n+1] means remainder when 7(3n+1) = 21n + 7 is divided by 3, we get remainder as 1. Hence [3n+1] = 1


(21n + 7)/3 = 21n/3 + 7/3 , we get remainder as 0 for 21n/3 & 1 for 7/3, hence final remainder is (0 + 1) = 1

I think it is called Remainder theorem, which states if \(R_1\) , \(R_2\) & \(R_3\) are remainders when \(N_1\), \(N_2\) & \(N_3\) are each divided by D

then,\(\frac{(N_1 + N_2 - N_3)}{D}\) the remainder will be \(R_1 + R_2 - R_3\)

I hope it is helpful.

Quote:
thanks for taking time to explain :-)
have a great weekend :)


No Problem! Have a good one!

Cheers!
GyM
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