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# How many two-digit whole numbers yield a remainder of 1 when divided

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Math Expert
Joined: 02 Sep 2009
Posts: 43867
How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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19 Jan 2015, 04:25
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35% (medium)

Question Stats:

64% (00:54) correct 36% (01:00) wrong based on 229 sessions

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How many two-digit whole numbers yield a remainder of 1 when divided by 10 and also yield a remainder of 1 when divided by 6?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Joined: 02 May 2014
Posts: 116
Schools: ESADE '16, HKU'16, SMU '16
GMAT 1: 620 Q46 V30
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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19 Jan 2015, 05:00
1
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2
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Bunuel wrote:
How many two-digit whole numbers yield a remainder of 1 when divided by 10 and also yield a remainder of 1 when divided by 6?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

The possible number N can be written as follow:
N = Multiple of LCM(6,10) + 1st such number
N = 30x + 1
Possible values = 1, 31, 61, 91
Answer : 3 such 2 digit number. D.
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Joined: 31 Jul 2014
Posts: 143
GMAT 1: 630 Q48 V29
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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19 Jan 2015, 05:01
1
KUDOS
Bunuel wrote:
How many two-digit whole numbers yield a remainder of 1 when divided by 10 and also yield a remainder of 1 when divided by 6?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

n=10p+1 --> Number could be 11 21 31 41 51
n=6q+1 --> Number could be 7 13 19 25 31

n= 30q+31
so n could be 31,61,91

IMO D
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Location: India
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Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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20 Jan 2015, 02:13
1
KUDOS

LCM of 10 & 6 = 30

Two-digit numbers giving remainder 1 for 30 are

31, 61, 91
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SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1839
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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20 Jan 2015, 02:21
Bunuel wrote:
How many two-digit whole numbers yield a remainder of 1 when divided by 10 and also yield a remainder of 1 when divided by 6?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

n=10p+1 --> Number could be 11 21 31 41 51
n=6q+1 --> Number could be 7 13 19 25 31

n= 30q+31
so n could be 31,61,91

IMO D

Can you explain the highlighted calculation? How is that obtained?
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Math Expert
Joined: 02 Sep 2009
Posts: 43867
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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20 Jan 2015, 02:31
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PareshGmat wrote:
Bunuel wrote:
How many two-digit whole numbers yield a remainder of 1 when divided by 10 and also yield a remainder of 1 when divided by 6?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

n=10p+1 --> Number could be 11 21 31 41 51
n=6q+1 --> Number could be 7 13 19 25 31

n= 30q+31
so n could be 31,61,91

IMO D

Can you explain the highlighted calculation? How is that obtained?

Positive integer n is divided by 10, the remainder is 1 --> $$n=10q+1$$, where $$q$$ is the quotient --> 1, 11, 21, 31, 41, ...
Positive integer n is divided by 6, the remainder is 1 --> $$n=6p+1$$, where $$p$$ is the quotient --> 1, 7, 13, 19, ...

There is a way to derive general formula for $$n$$ (of a type $$n=mx+r$$, where $$x$$ is divisor and $$r$$ is a remainder) based on above two statements:

Divisor $$x$$ would be the least common multiple of above two divisors 10 and 6, hence $$x=30$$.

Remainder $$r$$ would be the first common integer in above two patterns, hence $$r=1$$.

Therefore general formula based on both statements is $$n=30m+1$$. Thus n could be 1, 31, 61, 91, ... Since n is a two-digit integer, then n could only be 31, 61, or 91.

Check for more here: positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html#p721341

Hope it helps.
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Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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20 Jan 2015, 03:44
Find the least common factor and multiples of the number +1
Least common factor of 10 and 6 is 30 (two digits multiples of 30 are 30,60,90.. Add +1 to the numbers) so totally 3 numbers are possible
Intern
Joined: 08 Dec 2013
Posts: 41
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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15 Mar 2015, 20:47
hi Bunuel
dont you think with respect to your answer..
since q is the quotient..how can you put q=0 and get 1 as common from both equations
I mean if you put q=0,then n=1 but n is a two digit number so the first common value needs to be 31 i.e N(two digit)=30m+31..
thanks
Math Expert
Joined: 02 Sep 2009
Posts: 43867
Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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15 Mar 2015, 21:34
shreygupta3192 wrote:
hi Bunuel
since q is the quotient..how can you put q=0 and get 1 as common from both equations
I mean if you put q=0,then n=1 but n is a two digit number so the first common value needs to be 31 i.e N(two digit)=30m+31..
thanks

I first found general formula and then applied the restriction.
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Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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25 Jun 2016, 20:11
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Re: How many two-digit whole numbers yield a remainder of 1 when divided [#permalink]

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16 Aug 2017, 08:26
31,61 & 91 are the only three two digit numbers that when divided by 10 and 6 each leaves a remainder of 1.
Option D.
Re: How many two-digit whole numbers yield a remainder of 1 when divided   [#permalink] 16 Aug 2017, 08:26
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