GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 24 May 2020, 19:04

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 64068
(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =  [#permalink]

### Show Tags

13 Nov 2014, 08:48
00:00

Difficulty:

5% (low)

Question Stats:

88% (01:12) correct 12% (02:56) wrong based on 79 sessions

### HideShow timer Statistics

Tough and Tricky questions: Arithmetic.

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

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

Kudos for a correct solution.

_________________
Manager
Joined: 21 Jul 2014
Posts: 117
Re: (5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =  [#permalink]

### Show Tags

13 Nov 2014, 09:35
1
1
Bunuel wrote:

Tough and Tricky questions: Arithmetic.

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

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

Kudos for a correct solution.

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).

SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1709
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: (5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =  [#permalink]

### Show Tags

13 Nov 2014, 19:40
1
Answer = A = $$5^{16} - 1$$

Just add the powers of 5

$$(5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) = 5^{(8+4+2+2)} - 1 = 5^{16} - 1$$
Intern
Joined: 04 Jul 2014
Posts: 43
Schools: Smeal" 20
Re: (5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =  [#permalink]

### Show Tags

14 Nov 2014, 04:12
1
Bunuel wrote:

Tough and Tricky questions: Arithmetic.

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

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

Kudos for a correct solution.

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)

Math Expert
Joined: 02 Sep 2009
Posts: 64068
Re: (5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =  [#permalink]

### Show Tags

14 Nov 2014, 08:24
Bunuel wrote:

Tough and Tricky questions: Arithmetic.

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

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

Kudos for a correct solution.

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: $$(5^8 + 1)(5^4 + 1)(5^4 - 1) = (5^8 + 1)(5^8 - 1) = (5^16 - 1)$$.

_________________
Re: (5^8 + 1)(5^4 + 1)(5^2 + 1)(5^2 - 1) =   [#permalink] 14 Nov 2014, 08:24