(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =

Question

Tough and Tricky questions: Arithmetic.

A.  5^{16} - 1 
B.  5^{16} + 1 
C.  5^{32} - 1 
D.  5^{128} - 1 
E.  5^{16}(5^{16} - 1)

Kudos for a correct  solution .

LighthousePrep · Answer

Work shown below for the easy solution. As a general rule, it is always good to look at the patterns in the problem when it looks like there might be a lot of tedious math involved.

Skills you need to develop for this problem: understand exponent properties, multiplying equations (and factoring equations).

https://i.imgur.com/HZQKbjZ.jpg

PareshGmat · Answer

Answer = A =  5^{16} - 1

Just add the powers of 5

(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) = 5^{(8+4+2+2)} - 1 = 5^{16} - 1

Vetrik · Answer

The last two terms (5^2 + 1) (5^2 - 1) are of the form (a - b) (a + b). So it can be converted to the form (a^2 - b^2).

So now the 3rd term is ( (5^2)^2 - (1)^2) = (5^4 - 1)

Similarly 2nd and 3rd term becomes (5^8 - 1)

Now the first two becomes (5^16 - 1)

Hence Answer 'A'

Bunuel · Answer

Official Solution:

(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =

A.  5^{16} - 1 
B.  5^{16} + 1 
C.  5^{32} - 1 
D.  5^{128} - 1 
E.  5^{16}(5^{16} - 1)

The question asks us to simplify an expression.

We do not need to use FOIL to multiply the terms out. Instead, notice that  (5^2 + 1)(5^2 - 1)  is in the form  (x + y)(x - y) , which is the difference of two squares:  (x + y)(x - y) = x^2 - y^2 .

Since  (5^2 + 1)(5^2 - 1) = 5^4 - 1 , the original expression becomes:  (5^8 + 1)(5^4 + 1)(5^4 - 1) .

Another difference of squares has appeared:  (5^4 + 1)(5^4 - 1) .

We use the same method to get the final answer:  (5^8 + 1)(5^4 + 1)(5^4 - 1) = (5^8 + 1)(5^8 - 1) = (5^16 - 1) .

Answer: A.

Sfiso91 · Answer

Determine the units digits of the product (5^2018+1)×(5^2017+1)×(5^2016+1)……(5^1+1).

bumpbot · Answer

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(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =

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(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =

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