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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # How many such positive integers exist so that if the unit digit of the

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Manager  G
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]

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3
12 00:00

Difficulty:   65% (hard)

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

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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  B
Joined: 02 Jun 2017
Posts: 1
Re: How many such positive integers exist so that if the unit digit of the  [#permalink]

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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  P
Joined: 07 Dec 2014
Posts: 1230
How many such positive integers exist so that if the unit digit of the  [#permalink]

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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  S
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]

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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  B
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]

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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 V
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]

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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]

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_________________ Re: How many such positive integers exist so that if the unit digit of the   [#permalink] 10 Oct 2019, 13:03
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