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

 It is currently 17 Oct 2019, 13:36

### 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

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

# If a is the units digit of 7^47

Author Message
TAGS:

### Hide Tags

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3074
If a is the units digit of 7^47  [#permalink]

### Show Tags

Updated on: 13 Aug 2018, 02:49
1
16
00:00

Difficulty:

95% (hard)

Question Stats:

45% (02:13) correct 55% (02:32) wrong based on 179 sessions

### HideShow timer Statistics

e-GMAT Question:

If $$a$$ is the units digit of $$7^{47}$$ and b is the rightmost nonzero digit in $$(125^{10}× 28^{15})$$. What is the value of $$a+b$$?
A) 1
B) 2
C) 5
D) 6
F) 8

This is

Question 6 of The e-GMAT Number Properties Marathon

Go to

Question 7 of the Marathon

_________________

Originally posted by EgmatQuantExpert on 28 Feb 2018, 03:00.
Last edited by EgmatQuantExpert on 13 Aug 2018, 02:49, edited 2 times in total.
Current Student
Joined: 07 Jan 2016
Posts: 1088
Location: India
GMAT 1: 710 Q49 V36
Re: If a is the units digit of 7^47  [#permalink]

### Show Tags

28 Feb 2018, 03:15
1
1
EgmatQuantExpert wrote:

Question:

If $$a$$ is the units digit of $$7^{47}$$ and b is the rightmost nonzero digit in $$(125^{10}× 28^{15})$$. What is the value of $$a+b$$?
A) 1
B) 2
C) 5
D) 6
F) 8

a= unit digit of 7^47
7 has the unit cycle of 4

so 47 mod 4 = 3
and 7^3 = 343
unit digit 3

a=3

b = right most nonzero digit of 125 ^ 10 x 28 ^ 15

125 = 5^3

125^10 = 5^30

28 = 7 x 4

28^15 = 7^15 x 4^15
4 = 2^2

28^15 = 7^15 x 2^30

now $$(125^{10}× 28^{15})$$ = 5^30 x 7^15 x 2^30

we know 5x2=10 and the the powers are common so multiply the bases

$$(125^{10}× 28^{15})$$ = 10^30 x 7^15

for the right most non digit integer

we need 7^15 unit digit

we know 7 has cycle of 4
15 mod 4 = 3

so 7^15 will also end in 3 ( 7^3 = 343 unit digit 3 )

b=3

a+b= 3+3
6

(D) imo
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3335
Location: India
GPA: 3.12
Re: If a is the units digit of 7^47  [#permalink]

### Show Tags

28 Feb 2018, 03:33
1
1
EgmatQuantExpert wrote:

Question:

If $$a$$ is the units digit of $$7^{47}$$ and b is the rightmost nonzero digit in $$(125^{10}× 28^{15})$$. What is the value of $$a+b$$?
A) 1
B) 2
C) 5
D) 6
F) 8

The cyclicity of number 2 and 7 will be needed to solve the question

-n--------1---2---3---4
-2--------2---4---8---6
-7--------7---9---3---1

Since a is the units digit of $$7^{47}$$, it will be the units digit of $$7^{4*11 + 3}$$ or $$7^3$$, which is 3.

Similarly, b is the units digit of $$(125^{10}× 28^{15})$$

$$(125^{10} * 28^{15})$$ = $$(5^3)^{10} * (2^2 * 7)^{15}$$ = $$5^{30}*2^{30}*7^{4*3 + 3}$$ = $$10^{30} * 7^{4*3 + 3}$$ as $$a^m * b^ m = (ab)^m$$

This will translate to the units digit of $$7^3$$ which is 3.

Therefore, the value of a+b is 6(Option D)
_________________
You've got what it takes, but it will take everything you've got
Manager
Joined: 22 Jan 2014
Posts: 170
WE: Project Management (Computer Hardware)
Re: If a is the units digit of 7^47  [#permalink]

### Show Tags

28 Feb 2018, 05:15
EgmatQuantExpert wrote:

Question:

If $$a$$ is the units digit of $$7^{47}$$ and b is the rightmost nonzero digit in $$(125^{10}× 28^{15})$$. What is the value of $$a+b$$?
A) 1
B) 2
C) 5
D) 6
F) 8

7^1 ends in 7
7^2 ends in 9
7^3 ends in 3
7^4 ends in 1
then it repeates
so, 7^47 will have units digit same as 7^3 --> 3 = a

125^10 * 28^15 = 5^30 * 2^30 * 7^14 = 7*14 * 10^30 which will have 30 zeros preceded by units digit of 7^14 (same as that of 7^3 = 3 = b)

hence, a+b=6 (option d).
_________________
Illegitimi non carborundum.
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3074
Re: If a is the units digit of 7^47  [#permalink]

### Show Tags

28 Feb 2018, 12:12

Solution:

We need to the find the sum of $$a$$ and $$b$$ where, $$a$$ is the units digit of $$7^{47}$$ and $$b$$ is the rightmost nonzero digit in $$125^{10}× 28^{15}$$.
Let’s now calculate $$a$$and $$b$$.

Units digit of$$7^{47}$$
$$47= 4*11+3$$
$$4*11+3= 4k+3$$
Units digit of $$7^(4k+3)$$ is $$3$$
.
Thus, units digit of $$7^{47}$$ is $$3$$.
$$a= 3$$
Rightmost nonzero digit in 125^{10}× 28^{15}
125^{10}× 28^{15}= (5^3)^10× (2^2*7)^15.
125^{10}× 28^{15}= 5^30 ×2^30 ×7^15
125^{10}× 28^{15}= 10^30 ×7^15
We know, 10^n only gives n number of zeroes at the end of the number.
Hence, the units digit of 7^15 will be same as the rightmost nonzero digit of 125^{10}× 28^{15}.
$$15= 4*3+3$$
$$4*3+3= 4k+3$$
Units digit of $$7^(4k+3)$$ is $$3$$.
Thus,
$$b= 3$$
Hence,
$$a+b= 3+3$$
$$a+b=6$$
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13244
Re: If a is the units digit of 7^47  [#permalink]

### Show Tags

06 May 2019, 02:48
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If a is the units digit of 7^47   [#permalink] 06 May 2019, 02:48
Display posts from previous: Sort by