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

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'

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.

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