If the positive integer n leaves a remainder of 3 when divided by 7

Question

If the positive integer n leaves a remainder of 3 when divided by 7, which of the following statements would be true?

I. 4n + 2 is divisible by 14
II. n^2 - 2 is divisible by 7
III. (n + 3)(n + 4) is divisible by 7

A. Only I
B. Only II
C. Only III
D. I and II
E. I, II, and III

This is a  PS Butler Question

Check the links to other Butler Projects:

Data Sufficiency Butler  
  Critical Reasoning Butler  
  Reading Comprehension Butler  
  Data Insights Butler

ManifestDreamMBA · Answer

n = 7k+3

I: 4n + 2 = 4(7k+3) + 2 = 28k + 12 + 2 = 28k+14 = 14(2k+1)
This is divisible by 14
True
Eliminates B,C

II: n^2 - 2 = (7k+3)^2 - 2 = 49k^2 + 42k + 9 - 2 = 49k^2 + 42k + 7 = 7(7k^2 + 6k + 1)
This is divisible by 7
True
Eliminates A

III: (n + 3)(n + 4) = (7k+3 + 3)(7k+3 + 4) = (7k+6)(7k+7) = 7(k+1)(7k+6)
This is divisible by 7
True
Eliminates D

Answer E

Srrishti · Answer

from the question n = 7k+3 .where k is positive integer .
now lets take k =1 
n= 10 
4n+2 = 42 divisible by 14 
n^2-2 = 98 divisible by 7 
(n+3)(n+4) = 13*14 divisible by 7 
 hence answer is E since all conditions are true .

GMATinsight · Answer

the positive integer n leaves a remainder of 3 when divided by 7 may be {3, 10, 17..}

Let's take n = 3

I. 4n + 2 is divisible by 14, | 4*3+2 = 14 is divisible by 14
II. n^2 - 2 is divisible by 7 | 3^2-2 = 17 is divisible by 7
III. (n + 3)(n + 4) is divisible by 7 | (3+3)*(3+4) = 42 is divisible by 7

All TRUE hence

Answer: Option E

Dereno · Answer

Reading the question the number which came into my mind is 10.

Substituting 10 in the options,

I. 4n + 2 is divisible by 14

4*10+2 =42 , which is divisible by 14.

II. n^2 - 2 is divisible by 7

10^2 -2 = 98 , which is divisible by 7 .

III. (n + 3)(n + 4) is divisible by 7

(10+3)*(10+4) = 13*14 , which is divisible by 7. Hence  Option E.

But orthodox fool proof approach is :

let the number n = 7k +3 
substitute n in the options

I. 4n + 2 is divisible by 14

4*(7k+3) +2 = 28k+12+2 = 28k+14 =  14  *(2k+1) ———> which is of the format 14a. Hence divisible by 14.

II. n^2 - 2 is divisible by 7

(7k+3)^2 -2 = 49k^2+ 9+ 42k -2 = 49k^2 + 42k +7 =  7  *(7k^2+6k+1) . Divisible by 7.

III. (n + 3)(n + 4) is divisible by 7

(7k+6) (7k+7) =  7 *(k+1)*(7k+6) . Divisible by 7.

Hence,  Option E.

rohitz98 · Answer

From given the positive integer will be 7(a) + 3

I) 4n + 2 gives 28(a) + 14. Hence divisible by 14
II)n^2 - 2 gives 49(a) + 42(a) + 7. Hence divisible by 7
III) (n+3)(n+4) gives 7(a+1)(7a + 6). Hence divisible by 7

Hence  option E

Abhiswarup · Answer

n=7x+3
substituting values in 3 equations:
1. 4(7x+3)+2= 28x+14 Divisible by 14
2.(7x+3)^2-2= (49x^2+42x+7) Divisible by 7
3.(7x+6)(7x+7)again divisible by 7

All are correct. Answer is E.

If the positive integer n leaves a remainder of 3 when divided by 7

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the positive integer n leaves a remainder of 3 when divided by 7

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards