Aptitude - Numbers - Discussion

@Tanvi.

Here (a^3+b^3)identity is used. Please check it.

Let, (2^32) = (2^n)

If ( 2^n + 1 ) is divisible by a whole number, then higher powers of that same base will also be divisible by that same number.
(2^nm + 1 ) will also be divisible by the same number, where "m" is an integer.

Since, (2^96 + 1 ) = ( 2^32*3 + 1)
Therefore, (2^96 + 1) is divisible by the same number.
"Option D " is correct!

I used the logic by considering 32 as 2 and it's 3 times Multiple 96 as 6.

So, I used this value in this equation.

2^6+1/2^2+1.
=> 65/5 = 13.

Let 2^32 = x.

Lets take,
(2^96 + 1)
= [(2^32)3 + 1]
= (x^3 + 1) { we took 2^32=x}
=(x^3 + 1) { In the form of [a^3+b^3 ]=[a^2- ab+b^2 ]
= (x + 1)(x2 - x + 1).

Which is completely divisible by N, since (x + 1) is divisible by N.

Can anyone help me. What identity is used to determine whether a number is divisible or not completely?

Given: (2^32 + 1) is completely divisible by a whole number.
Let ( 2^32 + 1 ) = (2^n + 1 ), and since ( 2^n + 1 ) is divisible by a whole number,
we can use the exponentiation rule:

Formula:
( a^m )^n = a^(m * n).

Similarly,
(( 2^n )^ m + 1) is also divisible by the same whole number.
2^96 = ( ( 2^32)^3)+1.

Therefore, Option D: (2^96 + 1) is correct!

Thanks for the explanation.

Above mentioned are totally absurd. Few of them having tried good but not upto the mark.

Now see this carefully.

We are trying to get our given no. 2^32+1 from the choices given and I tried all d choices. And atlast 2^96+1 is easily broken to get 2^32+1.

Now I take an example to explain it in detail. Since 2^32+1 is completely div by N therefore N<=2^32+1 (less than equal to).

Ex: If 4 is completely div by x then x is always less than or equal to 4.

Now take first option 2^16+1 here it will be completely div N when N<=2^16+1 but wt if N>2^16+1 since value of N is ranging from 0 to 2^32+1 so wt if N>2^16+1 then 2^16+1 is not completely div.

Since we exactly don't know value of N we are checking its range to determine.

Therefore option A is not answer. Now similarly B is not answer not.

Option D is answer because all the range of N completely divides it which is explained how by indiabix.

If you don't understand this answer just try to focus on range of N. And again see my example of 4 completely divided.

@All. Please refer the following.

Formula : (a^3+b^3)=(a+b)(a^2-ab+b^2).
Substitute : (x^3+1^3)=(x+1)(x^2-x*(1)+1^2).

I'm not understanding this problem.