It is currently 19 Sep 2017, 15:45

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

What is the rightmost non-zero digit of 30^58 x 17^85?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Expert Post
1 KUDOS received
e-GMAT Representative
User avatar
S
Joined: 04 Jan 2015
Posts: 746

Kudos [?]: 2004 [1], given: 123

What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 30 Jul 2017, 13:08
1
This post received
KUDOS
Expert's post
3
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

33% (01:04) correct 67% (01:23) wrong based on 96 sessions

HideShow timer Statistics

What is the rightmost non-zero digit of \(30^{58}*17^{85}\)?

    A. 0
    B. 1
    C. 3
    D. 7
    E. 9



Thanks,
Saquib
Quant Expert
e-GMAT

Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts :)

Image
[Reveal] Spoiler: OA

_________________

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

Kudos [?]: 2004 [1], given: 123

Expert Post
e-GMAT Representative
User avatar
S
Joined: 04 Jan 2015
Posts: 746

Kudos [?]: 2004 [0], given: 123

Re: What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 30 Jul 2017, 13:09
Reserving this space to post the official solution. :)
_________________

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

Kudos [?]: 2004 [0], given: 123

1 KUDOS received
VP
VP
User avatar
D
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1005

Kudos [?]: 604 [1], given: 48

Location: Viet Nam
GMAT ToolKit User Premium Member CAT Tests
Re: What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 31 Jul 2017, 01:15
1
This post received
KUDOS
EgmatQuantExpert wrote:
What is the rightmost non-zero digit of \(30^{58}*17^{85}\)?

    A. 0
    B. 1
    C. 3
    D. 7
    E. 9


First, need to make a Prime Factorization.

\(30^{58}*17^{85} = (2*3*5)^{58} * 17 ^{85} = (2*5)^{58} * 3^{58} * 17 ^{85}\)

Note that \((2*5)^{58} = 10^{58}\) lead to zero digit. We just need to calculate the digit number of \(3^{58} * 17^{85}\)

Note that \(3^4=81=(...1)\).
Hence we have \(3^{58}=(3^4)^{14}*3^2=(...1)^{14} * 9 = (...9)\)

Note that \(17^2=(...9)\).
Hence we have \(17^4 = (...9)^2 = (...1) \implies 17^85=(17^4)^{21}*17=(...1)^{21} * 17 = (...7)\)

Hence \(3^{58} * 17^{85} = (...9) * (...7) = (...3)\)

The answer is C
_________________

Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer

Kudos [?]: 604 [1], given: 48

Director
Director
avatar
G
Joined: 22 May 2016
Posts: 623

Kudos [?]: 179 [0], given: 494

What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 01 Aug 2017, 16:27
EgmatQuantExpert wrote:
What is the rightmost non-zero digit of \(30^{58}*17^{85}\)?

    A. 0
    B. 1
    C. 3
    D. 7
    E. 9

Thanks,
Saquib
Quant Expert
e-GMAT

To find the rightmost non-zero digit, first get rid of zeros by factoring out any powers of 10.

Here the powers of 10 are in \(30^{58}\), which = \(3^{58}\) * \(10^{58}\). Factor out \(10^{58}\). The whole expression is now \(10^{58}\) * \((3^{58}*17^{85})\)

Ignore the powers of 10. Finding units digit involves only the units digit of the two numbers in parentheses: \(3^{58}\) and \(7^{85}\)

Units digit for both is determined by "cyclicity" of powers of 3 and 7:

\(3^1\) = 3 | \(7^1\) = 7
\(3^2\) = 9 | \(7^2\) = _9
\(3^3\) = _7| \(7^3\) = _3
\(3^4\) = _1| \(7^4\) = _1
---------------------------
\(3^5\) = _3| \(7^5\) = _7
\(3^6\) = _9| \(7^6\) = _9

After four powers, 3 and 7 repeat their patterns. Cyclicity is 4.

Units digit of \(3^{58}\)? Divide power by 4. (58/4) leaves a remainder of 2. The remainder determines where the 58th power "falls" in the cycle. Any power that leaves remainder of 2 has the same units digit as 3 to the power of 2. Units digit for \(3^{58}\) is same as \(3^2\), which is 9.

\(7^{85}\): 85/4 leaves remainder of 1. R1 has same units digit as \(7^1\), which is 7.

Finally, take the two units digits and multiply them together: 9*7 = 63. Rightmost non-zero digit is 3.

ANSWER C

Responding to an email: "rightmost" is literally the NON-zero digit "most to the right." Start from the first digit and move right until you get to the last ("rightmost") digit that is not zero.

Example: Imagine the number 987,650,000. The rightmost -----> non-zero digit here is [5]. Visually

987,6[5]0,000
------>[5]0,000

Hope it helps!

Kudos [?]: 179 [0], given: 494

Expert Post
Target Test Prep Representative
User avatar
S
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 1412

Kudos [?]: 753 [0], given: 5

Re: What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 01 Aug 2017, 17:09
EgmatQuantExpert wrote:
What is the rightmost non-zero digit of \(30^{58}*17^{85}\)?

    A. 0
    B. 1
    C. 3
    D. 7
    E. 9


Let’s simplify 30^58:

30^58 = 3^58 x 10^58

So, we have:

3^58 x 10^58 x 17^85

Since 10^58 is the number 1 followed by 58 zeros, we really need to determine the units digit of 3^58 and 17^85 (or 7^85):

We can evaluate 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 digits 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 have 1 as their units digit. Thus:

Following the pattern, we see that 3^56 has a units digit of 1, 3^57 has a units digit of 3, and 3^58 has a units digit of 9.

Next, we can evaluate 7^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are ONLY concerned with the units digit of 7 raised to each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of the units digits of powers of 7 repeats every 4 exponents. The pattern is 7–9–3–1. In this pattern, all positive exponents that are multiples of 4 will have 1 as their units digit. Thus:

7^84 has a units digit of 1 and 7^85 has a units digit of 7.

Since the units digit of 3^58 is 9 and the units digit of 7^85 is 7, and the product of 9 and 7 is 63, we know that the last non-zero digit of 30^58 and 17^85 is 3.

Answer: C
_________________

Jeffery Miller
Head of GMAT Instruction

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Kudos [?]: 753 [0], given: 5

Manager
Manager
avatar
B
Joined: 24 Jun 2017
Posts: 77

Kudos [?]: 13 [0], given: 120

Re: What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 01 Aug 2017, 22:13
broall wrote:
EgmatQuantExpert wrote:
What is the rightmost non-zero digit of \(30^{58}*17^{85}\)?

    A. 0
    B. 1
    C. 3
    D. 7
    E. 9


First, need to make a Prime Factorization.

\(30^{58}*17^{85} = (2*3*5)^{58} * 17 ^{85} = (2*5)^{58} * 3^{58} * 17 ^{85}\)

Note that \((2*5)^{58} = 10^{58}\) lead to zero digit. We just need to calculate the digit number of \(3^{58} * 17^{85}\)

Note that \(3^4=81=(...1)\).
Hence we have \(3^{58}=(3^4)^{14}*3^2=(...1)^{14} * 9 = (...9)\)

Note that \(17^2=(...9)\).
Hence we have \(17^4 = (...9)^2 = (...1) \implies 17^85=(17^4)^{21}*17=(...1)^{21} * 17 = (...7)\)

Hence \(3^{58} * 17^{85} = (...9) * (...7) = (...3)\)

The answer is C

Hi
why I cannot get the same response by simply calculating exponent cycles for primes (2∗3∗5)^58∗17^85
From my understanding you ignored 10^58 just for the sake of calculation simplicity as it gives 0

Kudos [?]: 13 [0], given: 120

Intern
Intern
avatar
B
Joined: 14 Oct 2016
Posts: 30

Kudos [?]: 6 [0], given: 147

Location: India
WE: Sales (Energy and Utilities)
Premium Member CAT Tests
Re: What is the rightmost non-zero digit of 30^58 x 17^85? [#permalink]

Show Tags

New post 13 Sep 2017, 11:58
Just to do calculations faster we can take this approach also


(10^58 ) ( 3^58) (17^85)

There is a formula a^n * b^n = (a*b)^n

so 85 can be written as 58 +27

( 3^58) (17^58) (17^27)

(51)^58 (17)^27

1^58 and 7^27 will give us the non zero digit

1 and 3

=3
_________________

Abhimanyu

Kudos [?]: 6 [0], given: 147

Re: What is the rightmost non-zero digit of 30^58 x 17^85?   [#permalink] 13 Sep 2017, 11:58
    Similar topics Author Replies Last post
Similar
Topics:
7 EXPERTS_POSTS_IN_THIS_TOPIC a, b, c, d and e are all non-zero distinct single digit integers. What ykaiim 5 27 Jun 2016, 21:51
2 EXPERTS_POSTS_IN_THIS_TOPIC If the units digit of x^3 is 6, what is the units digit of integer x? Bunuel 3 23 Jan 2017, 18:13
68 EXPERTS_POSTS_IN_THIS_TOPIC † and ¥ represent nonzero digits, and (†¥)² - (¥†)² WoundedTiger 13 08 Jul 2017, 12:33
6 EXPERTS_POSTS_IN_THIS_TOPIC What is the last non-zero digit of expression umg 5 05 Jun 2017, 10:29
18 What is the rightmost non-zero digit of 20!? sergbov123 26 27 Jan 2014, 10:06
Display posts from previous: Sort by

What is the rightmost non-zero digit of 30^58 x 17^85?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.