• Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep
  • Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • examPAL
    Most awarded test prep in the world
    Now free for 30 days

    Available with Beat the GMAT members only code

    MORE DETAILS
    examPAL
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider

In the product 2^19*9^13*5^24

This topic has 4 expert replies and 3 member replies

In the product 2^19*9^13*5^24

Post Thu Oct 12, 2017 6:21 am
In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9

What is the correct solution to this to get the right Option? How will i formulate the formula in this?

OA D

  • +1 Upvote Post
  • Quote
  • Flag
Top Reply
Post Thu Oct 12, 2017 2:38 pm
lheiannie07 wrote:
In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9

What is the correct solution to this to get the right Option? How will i formulate the formula in this?

OA D
Hi lheiannie07,
Let's take a look at your problem.
It seems a very interesting question.

2^19*9^13*5^24
Let's rewrite 5^24 as 5^19. 5^5

= (2^19)*(9^13)*(5^19)*(5^5)

Write the bases with the same exponents 19 together.
= (2^19)*(5^19)*(9^13)*(5^5)

We can write (2^19)*(5^19) as (2*5)^19
= ((2*5)^19)*(9^13)*(5^5)
= (10^19)*(9^13)*(5^5)

The term (10^19) indicates that this product has 19 zeros.
We are asked to find the digit at the 20th place to the left of the decimal point.
We already found out that there are 19 zeros to the left of the decimal point, so we need to find out the digit that comes right before the zeros.

To find out the digit right before the 19 zeros we need to look at the last digit of the product (9^13)*(5^5).
The last digit of 5^5 is a 5
Because when 5 is multiplied by itself odd number of times, the last digit of the product is always 5.
For example 5 * 1= 5, 5 *3 = 15, 5*5 = 25 etc

The last digit of 9^13 is an odd number
Because when 9 is multiplied by itself odd number of times, the last digit of the product is always odd.
For example. 9*1 = 9, 9*3 = 27, 9*5=45 etc

Now the last digit of 5^5 is a 5 and the last digit of 9^13 is an odd number, therefore the the last digit of the product of 5 and an odd number is again 5.

Therefore, the digit right before the 19 zeros will be the 5.

Hence, Option D is correct.
Hope this helps.

I am available if you'd like any follow up.

_________________
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

  • +1 Upvote Post
  • Quote
  • Flag
Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now.
Post Wed Jan 17, 2018 6:06 am
2^19*9^13*5^24 = 2^19*9^13*5^19*5^5 = (2*5)^19*9^13*5^5 = 9^13*5^5*10000000000000000000

As any odd multiple of 5 has 5 as its last digit, then the 20th digit is 5.

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2640 messages
Followed by:
113 members
Upvotes:
625
Target GMAT Score:
V51
GMAT Score:
780
Post Thu Oct 12, 2017 8:37 pm
It seems suspicious that we'd be asked for the 20th place, so let's see if there's some reason we'd get that easily.

Noticing that we've got 2¹⁹, we might think of 10¹⁹, which would get us to the 20th place pretty nicely. (If it's not clear why, see my post below.) And what do you know, there it is! =>

2¹⁹ * 5²⁴ =>

2¹⁹ * 5¹⁹ * 5⁵ =>

10¹⁹ * 5⁵

Now let's bring in our 9¹³:

10¹⁹ * 5⁵ * 9¹³

So our 20th digit will be the units digit of 5⁵ * 9¹³, which is just 5. (5 * anything odd ends in 5.)

  • +1 Upvote Post
  • Quote
  • Flag
Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2640 messages
Followed by:
113 members
Upvotes:
625
Target GMAT Score:
V51
GMAT Score:
780
Post Thu Oct 12, 2017 8:41 pm
If it's not clear why 10¹⁹ would determine the 20th place in our number above, take a look at few smaller examples.

Suppose we have 10² * 5. That'll give us 500, and the 5 becomes the THIRD digit from the right.

Suppose we have 10⁴ * 17. That'll give us 170,000, and the 7 (or the units digit of 17) is now the FIFTH digit from the right.

See the pattern? If we multiply by 10ⁿ, we shift everything n digits to the left, so the units digit of whatever we multiplied by will now be the (n + 1)th number from the right. Pretty cool! Smile

  • +1 Upvote Post
  • Quote
  • Flag
Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!
Post Wed Jan 10, 2018 12:29 am
Matt@VeritasPrep wrote:
If it's not clear why 10¹⁹ would determine the 20th place in our number above, take a look at few smaller examples.

Suppose we have 10² * 5. That'll give us 500, and the 5 becomes the THIRD digit from the right.

Suppose we have 10⁴ * 17. That'll give us 170,000, and the 7 (or the units digit of 17) is now the FIFTH digit from the right.

See the pattern? If we multiply by 10ⁿ, we shift everything n digits to the left, so the units digit of whatever we multiplied by will now be the (n + 1)th number from the right. Pretty cool! Smile
Thanks a lot!

  • +1 Upvote Post
  • Quote
  • Flag
Post Tue Jan 16, 2018 4:59 pm
lheiannie07 wrote:
In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9
Re-expressing 5^24 as 5^19 x 5^5, we can simplify the given expression:

2^19*9^13*5^24 = 2^19 x 5^19 x 9^13 x 5^5

Now, we combine 2^19 with 5^19, obtaining:

9^13 x 5^5 x 10^19

Recall that any number of the form m x 10^n (where m and n are positive integers) is the number m followed by n zeros. Since 9 raised to any power will always be odd, and since 5^5 will always end in a 5, 9^13 x 5^5 will end in a 5. Thus, the product 9^13 x 5^5 x 10^19 is a number ends with 19 zeros with the first nonzero digit to the left of these 19 zeros being a 5, which is also the 20th place to the the left of the decimal point.

Answer: D

_________________
Scott Woodbury-Stewart Founder and CEO

  • +1 Upvote Post
  • Quote
  • Flag
Post Wed Jan 17, 2018 6:08 am
If we can pair up multiples of 5 and 2, then each pair is a product of 10. The number of pairings is 19, so the last 19 places are zero.
The remaining factors are an odd number of 5s, resulting in the 20th digit being 5.

  • +1 Upvote Post
  • Quote
  • Flag

Best Conversation Starters

1 lheiannie07 112 topics
2 ardz24 71 topics
3 Roland2rule 69 topics
4 LUANDATO 53 topics
5 swerve 45 topics
See More Top Beat The GMAT Members...

Most Active Experts

1 image description GMATGuruNY

The Princeton Review Teacher

154 posts
2 image description Rich.C@EMPOWERgma...

EMPOWERgmat

107 posts
3 image description Jeff@TargetTestPrep

Target Test Prep

106 posts
4 image description Scott@TargetTestPrep

Target Test Prep

98 posts
5 image description EconomistGMATTutor

The Economist GMAT Tutor

91 posts
See More Top Beat The GMAT Experts