• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to \$200

Available with Beat the GMAT members only code

• Reach higher with Artificial Intelligence. Guaranteed
Now free for 30 days

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Free Trial & Practice Exam
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

## exponent problem

This topic has 1 expert reply and 6 member replies
ash_maverick Senior | Next Rank: 100 Posts
Joined
07 Jan 2007
Posted:
31 messages

#### exponent problem

Sun Jan 21, 2007 4:43 am
What is the remainder when 43^43+ 33^33 is divided by 10?

aim-wsc Legendary Member
Joined
20 Apr 2006
Posted:
2470 messages
Followed by:
14 members
85
Target GMAT Score:
801-
Fri Feb 02, 2007 9:14 pm
thats my suicide note none but this problem is responsible for it.

haha i know such big numbers are justbubbles. i dealt with units 5 6 1 they are of course easy. but this with 3 as a unit must belong to ''tough'' category.

_________________
Getting started @BTG?

Please do not PM me, (not active anymore) contact Eric.

limits660 Senior | Next Rank: 100 Posts
Joined
12 Jul 2006
Posted:
57 messages
Followed by:
2 members
Sun Feb 04, 2007 6:36 am
Stacey Koprince wrote:
Whenever they ask you a "units digit" question with really high exponents, there will be a pattern and you will only have to follow the units digit through the problem (b/c it is all multiplication).

43^43:

3^1 = 3^2 = 3^3 = 23^4 = 83^5 = etc. so the units digit pattern is 3-9-7-1. The pattern repeats every 4th term. So 3^4, 3^8, 3^12, etc, will all have the units digit 1. 3^40 will be 1, 3^41 will be 3, 3^42 will be 9, 3^43 will be 7. Same pattern as above. 3^32 will be 1, 3^33 will be 3. units digit 7 + units digit 3 will equal units digit 0. Anything that ends in 0 will have a remainder of 0 when divided by 10.
WOW, beautiful trick. I totally missed that one 8) Thanks

_________________
-
Jeff Sacco
www.jeffsacco.ca

chandra_adesh Newbie | Next Rank: 10 Posts
Joined
21 Jan 2007
Posted:
2 messages
Sun Jan 21, 2007 8:21 am
Last digit in the expansion of 43^43=7
Last digit in the expansion of 33^33=3

Last digit of 43^43+33^33=0

Hence remainder=0

g2000 Junior | Next Rank: 30 Posts
Joined
09 Jan 2007
Posted:
19 messages
1
Sun Jan 21, 2007 3:35 pm
i'm curious if it can be solved like below.
43^43 + 33^33 mod 10

Modular Exponentiation
X^a (mod n).

43^43 mod 10
First,
43^1 mod 10 = 3
43^2 mod 10 = 3^2 mod 10 = 9
43^4 mod 10 = 9^2 mod 10 = 1
43^8 mod 10 = 1^2 mod 10 = 1
....
43^32 mod 10 = 1 mod 10 = 1

43 = 32 + 8 + 2 + 1
43^43 mod 10
= 43^32 mod 10 * 43^8 mod 10 * 43^2 mod 10 + 43^1 mod 10
= (1 * 1 * 9 * 3 ) mod 10
= 27 mod 10
= 7

Same procedure is done on 33
33^1 mod 10 = 3
33^2 mod 10 = 3^2 mod 10 = 9
33^4 mod 10 = 9^2 mod 10 = 1
.....
33^32 mod 10 = 1 mod 10 = 1

33 = 32 + 1
33^33 mod 10
=33^32 mod 10 * 33^1 mod 10
= (1 * 3 ) mod 10
= 3 mod 10
= 3

Therefore, their sum must be 3 + 7 which gives the unit digit 0.
The remainder is obviously 0.

maxim730 Senior | Next Rank: 100 Posts
Joined
26 Dec 2006
Posted:
76 messages
Followed by:
2 members
Sun Jan 21, 2007 5:20 pm
Last digit in the expansion of 43^43=7
Last digit in the expansion of 33^33=3

Last digit of 43^43+33^33=0

Hence remainder=0
how did you get the last digit? 8)

### GMAT/MBA Expert

Stacey Koprince GMAT Instructor
Joined
27 Dec 2006
Posted:
2228 messages
Followed by:
681 members
639
GMAT Score:
780
Mon Jan 22, 2007 6:13 pm
Whenever they ask you a "units digit" question with really high exponents, there will be a pattern and you will only have to follow the units digit through the problem (b/c it is all multiplication).

43^43:

3^1 = 3^2 = 3^3 = 23^4 = 83^5 = etc. so the units digit pattern is 3-9-7-1. The pattern repeats every 4th term. So 3^4, 3^8, 3^12, etc, will all have the units digit 1. 3^40 will be 1, 3^41 will be 3, 3^42 will be 9, 3^43 will be 7. Same pattern as above. 3^32 will be 1, 3^33 will be 3. units digit 7 + units digit 3 will equal units digit 0. Anything that ends in 0 will have a remainder of 0 when divided by 10.

_________________
Please note: I do not use the Private Messaging system! I will not see any PMs that you send to me!!

Stacey Koprince
GMAT Instructor
Director of Online Community
Manhattan GMAT

Contributor to Beat The GMAT!

Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week.
telugupilla Newbie | Next Rank: 10 Posts
Joined
02 Feb 2007
Posted:
6 messages
Fri Feb 02, 2007 2:00 pm
Thanks for this cool tip!

### Best Conversation Starters

1 lheiannie07 80 topics
2 LUANDATO 59 topics
3 ardz24 52 topics
4 AAPL 45 topics
5 Roland2rule 43 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Rich.C@EMPOWERgma...

EMPOWERgmat

133 posts
2 Brent@GMATPrepNow

GMAT Prep Now Teacher

131 posts
3 GMATGuruNY

The Princeton Review Teacher

130 posts
4 Scott@TargetTestPrep

Target Test Prep

118 posts
5 Jeff@TargetTestPrep

Target Test Prep

114 posts
See More Top Beat The GMAT Experts