If 4^16 + 1 is divisible by positive integer x, then which of the foll

Question

If  4^{16} + 1  is divisible by positive integer x, then which of the following must also be divisible by x?

A.  4^{96} + 1

B.  4^{48} + 1

C.  4^{32} + 1

D.  4^{16} - 1

E.  4^8 + 1

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Kinshook · Accepted Answer

y^3 + 1 = (y+1)(y^2 - y +1)

If (y+1) is divisible by x - >( y^3 + 1) will also be divisible by x

4^{16*3} + 1 = 4^48 + 1 will also be divisible by x

IMO B

gurmukh · Answer

a^3 +b^3 =(a+b)(a^2-ab+b^2)

4^48+1= (4^16)^3 +1^3 = (4^16+1)× some integer

Therefore option B is also divisible by x

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Kritisood · Answer

If one doesnt remember the formula for a^3+b^3 is there another approach?

gurmukh · Answer

The point is not to know remember the formula because that anyway you can multiply and get the answer. The point is whether you can view the sum from that angle or not. If you remember the value of (a^3+b^3) then that helps you to develop the approach to solve the sum.

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exc4libur · Answer

(4^(16))^3+(1)^3=(a+b)(a^2+b^2-ab)
(4^(16)+1)((4^(16))^2+1-4^(16))
a+b is div by x

ans (B)

gurmukh · Answer

Check your calculation, there is mistake in it..

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supremekai · Answer

One approach if you don't remember the a^3 - b^3 formula :-

4^{16} + 1 = mx  (some multiple of x)
 4^{16} = mx - 1

Now, let's check the options:

For option B:  4^{48} + 1 = 4^{16 	imes 3} + 1 = (mx - 1)^3 + 1

Expanding  (mx - 1)^3 , we get:
 (mx - 1)^3 = m^3x^3 - 3m^2x^2 + 3mx - 1

So,  (mx - 1)^3 + 1 = m^3x^3 - 3m^2x^2 + 3mx - 1 + 1 = m^3x^3 - 3m^2x^2 + 3mx

This is clearly a multiple of x, so  4^{48} + 1  must also be divisible by x.

Therefore, the correct answer is B:  4^{48} + 1 .

If 4^16 + 1 is divisible by positive integer x, then which of the foll

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GMAT Problem Solving (PS) Questions

If 4^16 + 1 is divisible by positive integer x, then which of the foll

Prep Toolkit

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