What is the remainder when 7^8 is divided by 100?

This topic has expert replies
User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

[Math Revolution GMAT math practice question]

What is the remainder when 7^8 is divided by 100?

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

User avatar
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

by GMATGuruNY » Thu Jul 26, 2018 2:30 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the remainder when 7^8 is divided by 100?

A. 1
B. 2
C. 3
D. 4
E. 5
When an integer is divided by 100, the remainder will have the same units digit as the integer.
Thus, to determine which answer choice represents the remainder when 7� is divided by 100, we need to know the units digit of 7�.
When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7� --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, 7� has a units digit of 1.

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

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Thu Jul 26, 2018 5:13 am
Max@Math Revolution wrote: What is the remainder when 7^8 is divided by 100?

A. 1
B. 2
C. 3
D. 4
E. 5
Let's examine 7^8 - 1

Why would I do this?
Well, I know that 7^2 + 1 = 50, which is a factor of 100.
So, perhaps it's the case that 7^8 - 1 is divisible by 100, in which case 7^8 will leave a remainder of 1 when divided by 100

7^8 - 1 is a difference of squares.
So, 7^8 - 1 = (7^4 + 1)(7^4 - 1)
= (7^4 + 1)(7^2 + 1)(7^2 - 1)
= (7^4 + 1)(7^2 + 1)(7 + 1)(7 - 1)
= (7^4 + 1)(50)(8)(6)
= (7^4 + 1)(2400)
= (7^4 + 1)(24)(100)

So, we can see that 7^8 - 1 is divisible by 100
7^8 is 1 greater than 7^8 - 1, so we must get a remainder of 1 when 7^8 is divided by 100

Answer: A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image

User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

by Max@Math Revolution » Sun Jul 29, 2018 5:21 pm
=>

The remainder when 7^8 is divided by 100 is equal to the final two digits of 7^8.
Now, 7^1 = 7, 7^2 = 49, 7^3 = 343, and 7^4 = 2401.
So, the final two digits of 7^n have period 4:
The tens digits are 0 -> 4 -> 4 -> 0
and the units digits are 7 -> 9 -> 3 -> 1.
It follows that the tens and units digits of 7^8 are 0 and 1, respectively.
Therefore, the remainder when 7^8 is divided by 100 is 1.
Therefore, the answer is A.
Answer : A

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Mon Jul 30, 2018 10:27 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the remainder when 7^8 is divided by 100?

A. 1
B. 2
C. 3
D. 4
E. 5
Since 7^4 = (7^2)^2 = 49^2 = 2401, we see that when 7^4, or 2401, is divided by 100, the remainder is 1. Since 7^8 = (7^4)^2, the remainder when 7^8 is divided by 100 will be the same as the square of the remainder when 7^4 is divided by 100. Therefore, that remainder is 1^2 = 1.

Answer: A

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews