GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 27 Jan 2020, 02:22

### 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

# How many such positive integers exist so that if the unit digit of the

Author Message
TAGS:

### Hide Tags

Manager
Joined: 24 Dec 2016
Posts: 60
Location: United States
Concentration: Statistics
Schools: Duke Fuqua
GMAT 1: 720 Q49 V40
GPA: 3.38
How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

23 Jun 2017, 08:34
3
12
00:00

Difficulty:

65% (hard)

Question Stats:

59% (02:03) correct 41% (02:10) wrong based on 233 sessions

### HideShow timer Statistics

How many such positive integers exist so that if the unit digit of the original integer is removed, the ratio of the new number to the original one becomes $$\frac{1}{14}$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Intern
Joined: 02 Jun 2017
Posts: 1
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

23 Jun 2017, 09:37
5
1
4
You should start by translating the problem to symbols:

old= 10r + u (i.e. r=1 u = 3 so 10*1 +3 = 13)
new= r

$$\frac{new}{old} = \frac{1}{14}$$ so:

$$\frac{10r + u}{r} = \frac{1}{14}$$

which simplifies to: $$4r = u$$

now we can list posible solutions starting:

$$r=1 => u = 4$$
$$r=2 => u = 8$$
$$r=3 => u = 12$$ wrong because u is unit number so must be smaller or equal to 9

hence we have 2 solutions
##### General Discussion
VP
Joined: 07 Dec 2014
Posts: 1230
How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

23 Jun 2017, 11:57
Vardan95 wrote:
How many such positive integers exist so that if the unit digit of the original integer is removed, the ratio of the new number to the original one becomes $$\frac{1}{14}$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

let x=tens digit
y=units digit
(10x+y)/x=14
4x=y
y must be a one digit multiple of 4, either 4 or 8
x must be either 1 or 2
(10*1+4)/1=14
(20*1+8)/2=14
14 and 28 are 2 positive integers
C
Manager
Joined: 23 May 2017
Posts: 229
Concentration: Finance, Accounting
WE: Programming (Energy and Utilities)
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

23 Jun 2017, 12:16
The original number must be a multiple of 14..

14, 28, 42, 56, ....

let's start converting
= $$\frac{1}{14}$$( dropping 4 from 14) = $$\frac{1}{14}$$ = solution
= $$\frac{2}{28}$$( dropping 2 from 28) = $$\frac{1}{14}$$ = solution
= $$\frac{4}{42}$$( dropping 4 from 42) = $$\frac{2}{21}$$ = Not a solution
= $$\frac{5}{56}$$( dropping 5 from 56) = $$\frac{5}{56}$$ = Not a solution

so the denominator must be a multiple of numerator which is not possible as we proceed further : hence only 2 solution
Manager
Joined: 02 Feb 2016
Posts: 85
GMAT 1: 690 Q43 V41
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

10 Aug 2017, 14:17
lbowl wrote:
You should start by translating the problem to symbols:

old= 10r + u (i.e. r=1 u = 3 so 10*1 +3 = 13)
new= r

$$\frac{new}{old} = \frac{1}{14}$$ so:

$$\frac{10r + u}{r} = \frac{1}{14}$$

which simplifies to: $$4r = u$$

now we can list posible solutions starting:

$$r=1 => u = 4$$
$$r=2 => u = 8$$
$$r=3 => u = 12$$ wrong because u is unit number so must be smaller or equal to 9

hence we have 2 solutions

Haven't you reversed the old/new when applying the ratio?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10018
Location: Pune, India
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

10 Aug 2017, 23:45
1
1
Vardan95 wrote:
How many such positive integers exist so that if the unit digit of the original integer is removed, the ratio of the new number to the original one becomes $$\frac{1}{14}$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

The given ratio is 1/14. This is a solution. If the units digit of 14 is removed, we get 1. Let's look at the multiples of this ratio.
2/28 - Solution
3/42
4/56
5/70
6/84
7/98
8/112
...

Note the pattern. In the denominator, if you remove the units digit, the gap between what you get and the numerator is increasing. So you will not have another solution.

_________________
Karishma
Veritas Prep GMAT Instructor

Non-Human User
Joined: 09 Sep 2013
Posts: 14011
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

### Show Tags

10 Oct 2019, 13:03
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: How many such positive integers exist so that if the unit digit of the   [#permalink] 10 Oct 2019, 13:03
Display posts from previous: Sort by