GMAT Question of the Day - Daily to your Mailbox; hard ones only
close

It is currently 20 Aug 2019, 16:06
Register
GMAT Club Tests
Decision Tracker
My Rewards
New comers' posts
New posts
Unanswered

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Go to My Error Log Learn more
Close

Request Expert Reply

Confirm Cancel
CLICK HERE TO HIDE/SHOW EVENTS

What is the remainder when 5^16 - 3^16 is divided by 8?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
  Sort by Date Sort by Kudos  
Search for:
 
Print view
First unread post
Author Message
TAGS:

Hide Tags

Find Similar Topics 
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3019
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

Show Tags

New post 19 Dec 2018, 11:18
1
10
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

77% (01:42) correct 23% (02:12) wrong based on 143 sessions

HideShow timer Statistics

What is the remainder when \(5^{16} - 3^{16}\) is divided by 16?

    A. 0
    B. 1
    C. 3
    D. 5
    E. 7

To read all our articles:Must read articles to reach Q51

Image
Spoiler: OA

_________________







Register for free sessions
Number Properties:- Get 5 free video lessons, 50 practice questions | Algebra:-Get 4 free video lessons, 40 practice questions
Free Strategy Session:- Key strategy to score 760

Success Stories
Q38 to Q50- Interview call form Wharton | Q35 to Q50 | More Success Stories

Ace GMAT
Articles and Questions to reach Q51 | Question of the week | Tips From V40+ Scorers | V27 to V42:- GMAT 770 | Guide to Get into ISB-MBA

Must Read Articles
Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Signature Read More
VP
VP
User avatar
D
Joined: 31 Oct 2013
Posts: 1429
Concentration: Accounting, Finance
GPA: 3.68
WE: Analyst (Accounting)
CAT Tests
Re: What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

Show Tags

New post 19 Dec 2018, 11:50
2
EgmatQuantExpert wrote:
What is the remainder when \(5^{16} - 3^{16}\) is divided by 16?

    A. 0
    B. 1
    C. 3
    D. 5
    E. 7

To read all our articles:Must read articles to reach Q51

Image



\(5^{16} - 3^{16}\)

\(5^8*2 - 3^8*2\)

\((5^8 +3^8) (5^8 - 3^8)\)

now just break down 5^8 - 3^8.

apply \(a^2 -b^2.\)

at least we got : (\(5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3)\)

Now divide :

\((5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3) /16.\)

\((5^8 +3^8) (5^4 +3^4) (5^2 +3^2) *8*2 /16.\)

Reminder is 0.

A is the correct answer.
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 4491
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member CAT Tests
Re: What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

Show Tags

New post 20 Dec 2018, 01:25
2
I solved using cyclicity

for any power to 5 we would always get 5 as units digit and for 3 ^ 16 we would get 1 as unit digits

so 5-1= 4 unit digit

16= 2^4

for an even Nr ending with 4 unit digit when divided by 2 would always give remainder as 0 , so IMO A ..




EgmatQuantExpert wrote:
What is the remainder when \(5^{16} - 3^{16}\) is divided by 16?

    A. 0
    B. 1
    C. 3
    D. 5
    E. 7

To read all our articles:Must read articles to reach Q51

Image

_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
Signature Read More
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3019
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

Show Tags

New post 01 Jan 2019, 22:56

Solution


To find:
We are asked to find out,
    • The remainder when \(5^{16} – 3^{16}\) is divided by 8

Approach and Working:
    • \(5^{16} – 3^{16}\) can be written as \((5^8)^2 – (3^8)^2\)

We know that, \(a^2 – b^2 = (a + b) * (a - b)\)
    • Thus, \((5^8)^2 – (3^8)^2 = (5^8 + 3^8) * (5^8 – 3^8) = (5^8 + 3^8) * (5^4 + 3^4) * (5^4 – 3^4) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5 + 3) * (5 – 3)\)
    • So, \(5^{16} – 3^{16} = 16 * (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2)\)

If you observe carefully, the above expression is a multiple of 8
    • Therefore, the remainder will be 0

Hence the correct answer is Option A.

Answer: A

Image

_________________







Register for free sessions
Number Properties:- Get 5 free video lessons, 50 practice questions | Algebra:-Get 4 free video lessons, 40 practice questions
Free Strategy Session:- Key strategy to score 760

Success Stories
Q38 to Q50- Interview call form Wharton | Q35 to Q50 | More Success Stories

Ace GMAT
Articles and Questions to reach Q51 | Question of the week | Tips From V40+ Scorers | V27 to V42:- GMAT 770 | Guide to Get into ISB-MBA

Must Read Articles
Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Signature Read More
GMAT Club Bot
What is the remainder when 5^16 - 3^16 is divided by 8?   [#permalink] 01 Jan 2019, 22:56
Print view
First unread post
Display posts from previous: Sort by

What is the remainder when 5^16 - 3^16 is divided by 8?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Moderators: chetan2u, VeritasKarishma, Bunuel, Gladiator59, generis

Search for:





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Main navigation

GMAT Resources

Partners

MBA Resources

www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions