If a is the units digit of 7^47 : Problem Solving (PS)

My Profile Logout

Test's Subscription Expires:

My Follow Feed Kudos My Error Log

Messages and Replies

Settings Mark All Read

See All

Applicant Notifications

Settings Mark All Read

See All

Global notifications

Settings Mark All Read

See All

If a is the units digit of 7^47

Sort by Date

Sort by Kudos

Author

Message

TAGS:

Hide Tags

e-GMAT Representative

Joined: 04 Jan 2015

Posts: 2324

If a is the units digit of 7^47 [#permalink]

Show Tags

Updated on: 13 Aug 2018, 01:49

00:00

Difficulty:

85% (hard)

Question Stats:

46% (02:15) correct

54% (02:23) wrong

based on 121 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

Spoiler: OA

_________________

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Originally posted by EgmatQuantExpert on 28 Feb 2018, 02:00.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:49, edited 2 times in total.

BSchool Forum Moderator

Joined: 07 Jan 2016

Posts: 832

Location: India

GMAT 1: 710 Q49 V36

Re: If a is the units digit of 7^47 [#permalink]

Show Tags

28 Feb 2018, 02:15

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

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: 3327

Location: India

GPA: 3.12

Re: If a is the units digit of 7^47 [#permalink]

Show Tags

28 Feb 2018, 02:33

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: 174

WE: Project Management (Computer Hardware)

Re: If a is the units digit of 7^47 [#permalink]

Show Tags

28 Feb 2018, 04:15

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: 2324

Re: If a is the units digit of 7^47 [#permalink]

Show Tags

28 Feb 2018, 11:12

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\(7^{47}\)

.
Thus, units digit of \(7^{47}\) is \(3\).

\(a= 3\)Rightmost nonzero digit in 125^{10}× 28^{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}.

Thus,

\(b= 3\)Hence,

Answer: Option D
_________________