Aptitude - Numbers - Discussion

6n^2+6n= 6n(n+1).

Now the number above will always be divisible by 6
And it will be divisible by 12 only when n(n+1) part is even
Eg.6*2=12 ,6*6=36,6*5=30 when we used odd number to multiply with 6 its resultant was not divisible by 12 whereas with even number multiplies by 6 is always divisible by 12.

In case of n*(n+1) always one of the numbers will be odd and another one will be even, Multiplication of odd and even number always results into even number hence it will always be divisible by 12.

Firstly, we have to use 6n(n+1).

Where as n= natural number, we can use any number what you desire,
For eg 1, when n=2, 6*2(2+1) =12*3 =36,

36 can be divided by 6 , 12 and 18.

Eg 2, where n=1, 6*1(1+1)=6*2 =12.

12 can be divided by 6 and 12.

Check with given numbers,

A, C and D are not available. So, C is an answer. If there is an E which is non of these. We have to choose it to be exact b/c we can use any number in a place of n. If so, 6, 12 and 18 are available.

If a number is divisible by 12 or 18 then it is always divisible by 6.
So option C and D are eliminated...

We can see the no. 6n(n+1) is divisible by 6 so we have to check for 12
Now a number would be divisible by 12 if it is multiplication of 6 and a even number. (e.g. (6*even number)

Now n(n+1) is even number.
So answer is B

Assume n is a odd number then n-1 will be even and if odd number multiply with even it will be even. So that's for n(n-1) is a even number.

If you assume n is an even number then (n-1) will be odd number..Then multiplication of odd and even number will be also even number. So n(n-1) is always even.

Solve with prove
(6n2 + 6n).

1st
6n(n+1) put the value n=6.
66(6+1)= 66(7)=462/6=77 clearly divisible.

2nd
6n(n+1)= put the value n=12.
612(12+1)=612(13)=7956/12=633 clearly divisible.

Exception
6n(n+1)= put the value 18.
618(18+1)=618(19) = 11742/18 it's not divisible.

@Onkar @Narra Rakesh @Emi.

If n is odd then (n+1) is even and if n is even then (n+1) is odd.
Multiplication of an odd and an even number is always even.
So that the number is divisible by 6 and 12.

@Emi.

It mean that --** 6n (n+1) **--- where 6 is product & n may be even or odd.

That's why if product is even is always give even result whether -- n -- may have even or odd.

6n^2 + 6n = 6 n(n+1) ->n(n+1) is always even
= 6 2(2+1), Ex:2(2+1) = 6.
= 36.
->I think everyone knows divisibility of 6 and 12.

@All.

Let us assume n ( it can be any number)
I'm taking n as 1.
6n² + 6n = 6 × 1² + 6×1 = 12
So, 12 is divisible by both 6 and 12.
Answer: Option B

If N is even number given expression is divisible by 6 and 12.
But, here N is a natural number.
The given expression is divisible by 6 only.