Latest TCS Aptitude Question SOLUTION: The number of natural numbers n such that (n+1)^2/(n+7) is an integer, is i. 4 ii. 5 iii. 6 iv. None of these

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The number of natural numbers n such that (n+1)^2/(n+7) is an integer, is

i. 4
ii. 5
iii. 6
iv. None of these

Asked In TCS Tata Power (9 years ago)

Solved Sanu Panji Read Solution (5)

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Sanu Panji  (9 years ago) Best Solution

(n+1)^2/(n+7)
=(n^2+2n+1)/(n+7)

by simple algebraic polynomial division we get 36 as remainder
now (n+1)^2/(n+7) to be an integer 36 have to divisable by (n+7)
now 36 has 9 factors :1, 2, 3, 4, 6, 9, 12, 18 and 36
and only 4 factors are greater then 7 : 9, 12, 18, 36
so there are 4 natural numbers n such that (n+1)^2/(n+7) is an integer.
n =(9-7) = 2 or,
=(12-7)= 5 or,
=(18-7)= 11 or,
=(36-7)= 29
like:
when n=2
(n+1)^2/(n+7)=3^2/9=1 is an integer
when n=5
(n+1)^2/(n+7)=6^2/12=3 is an integer
when n=11
(n+1)^2/(n+7)=12^2/18=8 is an integer
when n=29
(n+1)^2/(n+7)=30^2/36=25 is an integer

so option (i) is correct
there are 4 such numbers

Is this solution Helpfull? Yes (32) | No (1)  

Pradeep  (9 years ago)

(n+1)^2/n+ 7 = k

k= (n+7)^2 /n+7 – (12n + 48)/n+7

if k is integer then (12 + 48)/n+7 has to be an integer.

12n +48/n+7=12(n+4)/n+7.

= 12 (n+7 -3)/n+7



36/n+7 has to be an integer.

n+7 = 9 ,12 , 18 , 36

n can take 4 values

Is this solution Helpfull? Yes (8) | No   

Koushik Das  (9 years ago)

Take the option and check one by one.
n=4 then (4+1)^2/(7+4)=2.27 is not an integer.
n=5 then (5+1)^2/(5+7)=3 is an integer.
n=6 then (6+1)^2/(6+7)=3.76 is not an integer.
So ans is ii. 5.

Is this solution Helpfull? Yes (4) | No (9)  

Sourabh Nath  (9 years ago)

we can use (n+1)^2 greater than (n+7) which gives n>2,-3........now there are four integers -2.-1,0,1 in between -3 and 2. so answer is 4.

Is this solution Helpfull? Yes (1) | No   

Arun kumar prajapati  (9 years ago)

Use optional checking .. Take n=5
Then (5+1)^2/(5+7)=36/12=3 I.e. an integer .
So ans is 5.

Is this solution Helpfull? Yes | No (3)  

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