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

It is currently 17 Oct 2018, 11:34

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.

Close

Request Expert Reply

Confirm Cancel

S98-01

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 49968
S98-01  [#permalink]

Show Tags

New post 16 Sep 2014, 01:52
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

45% (01:07) correct 55% (01:13) wrong based on 33 sessions

HideShow timer Statistics

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 49968
Re S98-01  [#permalink]

Show Tags

New post 16 Sep 2014, 01:52
Official Solution:

What is the remainder when you divide \(2^{200}\) by 7?

A. 1
B. 2
C. 3
D. 4
E. 5


We cannot compute \(2^{200}\) in anywhere near the time allotted, so we should look for a pattern in much simpler problems that we can scale up to \(2^{200}\).

The simpler problems we should solve are these:

What is the remainder when you divide 2 by 7?

What is the remainder when you divide \(2^2\) by 7?

What is the remainder when you divide \(2^3\) by 7?

What is the remainder when you divide \(2^4\) by 7?

... and so on with the powers of 2.

The answers follow this pattern:

2 divided by 7 leaves remainder 2.

\(2^2\) (which equals 4) divided by 7 leaves remainder 4.

\(2^3\) (which equals 8) divided by 7 leaves remainder 1.

\(2^4\) (which equals 16) divided by 7 leaves remainder 2.

\(2^5\) (which equals 32) divided by 7 leaves remainder 4.

\(2^6\) (which equals 64) divided by 7 leaves remainder 1.

We should stop as soon as we notice that the cycle will repeat itself forever in this pattern: [2, 4, 1]. Every third remainder is the same. (From here on out, "remainder" always means "remainder after we divide by 7.") Since every third remainder is the same, we should look at the remainder when the power is a multiple of 3. The remainders of 23 and 26 are 1. Thus, the remainder of 2 raised to a power that is any multiple of 3 is 1.

Now, 200 is not a multiple of 3, but we can look for a multiple of 3 near 200. 201 is a multiple of 3 (its digits add to 3), so \(2^{201}\) has a remainder of 1. Finally, we notice that the remainder of \(2^{200}\) must be one position earlier in the cycle than the remainder of \(2^{201}\). Since the cycle is [2, 4, 1], the remainder of \(2^{200}\) is 4.


Answer: D
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
B
Joined: 20 Aug 2016
Posts: 47
GMAT 1: 570 Q46 V23
GMAT 2: 610 Q49 V25
GMAT 3: 610 Q49 V25
WE: Information Technology (Other)
Reviews Badge
Re: S98-01  [#permalink]

Show Tags

New post 04 Oct 2018, 06:27
2^5 /7 gives 4 as remainder. Hence, power 200, which is multiple of 5, will give remainder as 4, as well
GMAT Club Bot
Re: S98-01 &nbs [#permalink] 04 Oct 2018, 06:27
Display posts from previous: Sort by

S98-01

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

Moderators: chetan2u, Bunuel



Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.