What is the largest positive integer n such that n2+7n+12/n2−n−12

Question

What is the largest positive integer n such that  \frac{n^2 + 7n + 12}{n^2−n−12}  is also positive integer?

A. 6
B. 8
C. 10
D. 12
E. 16

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chetan2u · Answer

Two ways.. 
 (I) Simplify and then substitute choices... 
 \frac{n^2 + 7n + 12}{n^2−n−12} ..
 =\frac{(n+4)(n+3)}{(n-4)(n+3)} ..
 =\frac{n+4}{n-4} ..

As we are looking for the largest value, start from E..
E) n=16.... \frac{n+4}{n-4}=\frac{16+4}{16-4}=\frac{20}{12} ..Not an integer
D) n=12.... \frac{n+4}{n-4}=\frac{12+4}{12-4}=\frac{16}{8}=2 ....YES

(II) Substitute the choices.. 
Again start from the largest
E) n=16..... \frac{n^2 + 7n + 12}{n^2−n−12} = \frac{16^2 + 7*16 + 12}{16^2−16−12}=\frac{256+112+12}{256-28}=\frac{380}{228} ...NO
D) n=12..... \frac{n^2 + 7n + 12}{n^2−n−12} = \frac{12^2 + 7*12 + 12}{12^2−12−12}=\frac{12(12+7+1)}{12(12-2)}=\frac{12*20}{12*10}=2 .....YES

D

ArunSharma12 · Answer

\frac{n^2 + 7n + 12}{n^2−n−12}  =  \frac{(n+4)(n+3)}{(n-4)(n+3)} 
 \frac{(n+4)}{(n-4)}  > 0
starting from option E,  \frac{(16+4)}{(16-4)} = \frac{(20)}{(12)} ; not an integer
 \frac{(12+4)}{(12-4)} = \frac{(16)}{(8)} =2; this is positive integer

Ans: D

nick1816 · Answer

\frac{n^2 + 7n + 12}{n^2−n−12}

=  \frac{(n+3)(n+4)}{(n+3)(n-4)}

=  \frac{(n+4)}{(n-4)}

=  \frac{(n-4+8)}{(n-4)}

=  1+ \frac{8}{n-4}

Since we want  \frac{n^2 + 7n + 12}{n^2−n−12}  to be an integer,  \frac{8}{n-4}  must be an integer.

Or n-4 must be a factor of 8.

Maximum positive value n-4 can take is 8.

n= 8+4=12

Kinshook · Answer

Asked: What is the largest positive integer n such that  \frac{n^2 + 7n + 12}{n^2−n−12}  is also positive integer?

(n+3)(n+4) / (n-4)(n+3) ; n is NOT = -3
(n+4)/(n-4) is an integer

A. 6
(6+4)/(6-4) = 10/2 = 5 is an integer
B. 8
(8+4)/(8-4) = 12/4 = 3 is an integer
C. 10
(10+4)/(10-4) = 14/6 is NOT an integer
D. 12
(12+4)/(12-4) = 16/8 = 2 is an integer
E. 16
(16+4)/(16-4) = 10/12 is not an integer

IMO D

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What is the largest positive integer n such that n2+7n+12/n2−n−12

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What is the largest positive integer n such that n2+7n+12/n2−n−12

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