If n = (33)^43 + (43)^33 what is the units digit of n?
A. 0
B. 2
C. 4
D. 6
E. 8
The OA is A.
How can I know the units digit without making the whole calculation? I need some help. Please.
If n = (33)^43 + (43)^33 what is the units digit of n?
This topic has expert replies
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
3¹ --> units digit of 3.M7MBA wrote:If n = (33)^43 + (43)^33 what is the units digit of n?
A. 0
B. 2
C. 4
D. 6
E. 8
3² --> units digit of 9. (Since the product of the preceding units digit and 3 = 3*3 = 9.)
3³ --> units digit of 7. (Since the product of the preceding units digit and 3 = 9*3 = 27.)
3� --> units digit of 1. (Since the product of the preceding units digit and 3 = 7*3 = 21.)
From here, the units digits will repeat in the same pattern: 3, 9, 7, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 3 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus:
33�� and 43³² each have a units digit of 1.
From here, the cycle of units digits will repeat: 3, 9, 7, 1...
Thus:
33�¹ and 43³³ each have a units digit of 3.
33�² has a units digit of 9.
33�³ has a units digit of 7.
Result:
Since n = 33�³ + 43³³, n ---> (units digit of 3) + (units digit of 7) = units digit of 0.
The correct answer is A.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
GMAT/MBA Expert
- Jeff@TargetTestPrep
- GMAT Instructor
- Posts: 1462
- Joined: Thu Apr 09, 2015 9:34 am
- Location: New York, NY
- Thanked: 39 times
- Followed by:22 members
Since we are asked to determine only the units digit of n, we can rewrite the expression as:M7MBA wrote:If n = (33)^43 + (43)^33 what is the units digit of n?
A. 0
B. 2
C. 4
D. 6
E. 8
n = 3^43 + 3^33
Let's now determine the units digit of 3^43 and 3^33
Let's start by evaluating the pattern of the units digits of 3^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 3. When writing out the pattern, notice that we are ONLY concerned with the units digit of 3 raised to each power.
3^1 = 3
3^2 = 9
3^3 = 7
3^4 = 1
3^5 = 3
The pattern of the units digit of powers of 3 repeats every 4 exponents. The pattern is 3-9-7-1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as its units digit. Thus:
3^44 has a units digit of 1.
Therefore, 3^43 has a units digit of 7.
and
3^32 has a units digit of 1.
Therefore, 3^33 has a units digit of 3.
Thus, n has a units digit of 0, since the units digit of 7 + 3 = 10 is 0.
Answer: A
Jeffrey Miller
Head of GMAT Instruction
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
GMAT/MBA Expert
- ceilidh.erickson
- GMAT Instructor
- Posts: 2095
- Joined: Tue Dec 04, 2012 3:22 pm
- Thanked: 1443 times
- Followed by:247 members
As Mitch described, finding the pattern in units digits is the key to solving this problem. For more practice on this concept, see:
https://www.beatthegmat.com/what-is-the ... tml#554073
https://www.beatthegmat.com/what-is-the ... tml#544267
https://www.beatthegmat.com/what-is-the ... tml#800962
https://www.beatthegmat.com/if-n-and-a- ... tml#784629
https://www.beatthegmat.com/what-is-the ... tml#554073
https://www.beatthegmat.com/what-is-the ... tml#544267
https://www.beatthegmat.com/what-is-the ... tml#800962
https://www.beatthegmat.com/if-n-and-a- ... tml#784629
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education