GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Nov 2019, 22:59 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # How many two-digit whole numbers yield a remainder of 1 when divided

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 59020
How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

2
15 00:00

Difficulty:   35% (medium)

Question Stats: 68% (01:20) correct 32% (01:23) wrong based on 503 sessions

### HideShow timer Statistics

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

Project PS Butler : Question #53

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 59020
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

4
10
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.
_________________
##### General Discussion
Manager  Joined: 02 May 2014
Posts: 91
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]

### Show Tags

1
2
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.
Manager  Joined: 31 Jul 2014
Posts: 125
GMAT 1: 630 Q48 V29 Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

1
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
SVP  Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1731
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]

### Show Tags

3

LCM of 10 & 6 = 30

Two-digit numbers giving remainder 1 for 30 are

31, 61, 91
_________________
Kindly press "+1 Kudos" to appreciate SVP  Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1731
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]

### Show Tags

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?
_________________
Kindly press "+1 Kudos" to appreciate Intern  Joined: 08 Jan 2015
Posts: 11
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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: 30
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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 V
Joined: 02 Sep 2009
Posts: 59020
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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.
_________________
Manager  B
Joined: 18 Jun 2017
Posts: 58
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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.
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15446
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

Hi All,

This type of question is rooted in pattern-matching. Once you find the patterns behind this question, it won't be hard to solve. Instead of trying to do every step all at once, I suggest that you break the prompt into "pieces":

First, name the 2-digit numbers that are evenly divisible by 10:

10, 20, 30, ......90

Now, name the 2-digit numbers that have a remainder of 1 when divided by 10:

11, 21, 31,.....91

Now that we've established the numbers that fit the first 2 "restrictions" in the prompt, let's factor in numbers that are ALSO divisible by 6:

30, 60, 90

And ALSO have a remainder of 1 when divided by 6:

31, 61, 91

GMAT assassins aren't born, they're made,
Rich
_________________
SVP  V
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 1701
Location: India
GPA: 3.01
WE: Engineering (Real Estate)
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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.

Two digit numbers divided my 10 yielding remainder 1 = 11,21,31,41,51,61,71,81,91

Two digit numbers divided my 6 yielding remainder 1 = 31,61,91

Three common numbers.

Hence (D)
_________________
"Do not watch clock; Do what it does. KEEP GOING."
Intern  B
Joined: 02 Oct 2016
Posts: 23
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

2
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.
Director  G
Joined: 09 Mar 2018
Posts: 994
Location: India
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

1
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

Project PS Butler : Question #53

We have to take the LCM and never the product

N1 = 10 k + 1
N2 = 6k + 1

N3 = 30 k + 1

k = 1,2,3

Only 3, two digit values can be yield.

D
_________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
Intern  Joined: 07 Oct 2018
Posts: 3
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

LCM (6,10) = 30
x = 10q+1 = 6r +1
=> x = 30m + 1 -> test m = 1,2,3 -> 31, 61,91
Director  V
Status: Manager
Joined: 27 Oct 2018
Posts: 718
Location: Egypt
GPA: 3.67
WE: Pharmaceuticals (Health Care)
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

Hi Bunuel
I have a small query regarding this question.

when I tried solving it, I included all positive and negative numbers assuming that the question didn't restrict to positive integers,
and I got -89,-59,-29,31,61,91 (which are six possible two digit numbers)

_________________
Thanks for Kudos
SVP  P
Joined: 03 Jun 2019
Posts: 1838
Location: India
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

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

Project PS Butler : Question #53

LCM(6,10) = 30
Numbers ={31,61,91}

IMO D

Posted from my mobile device
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
VP  D
Joined: 14 Feb 2017
Posts: 1273
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23 GMAT 3: 650 Q47 V33 GMAT 4: 650 Q44 V36 GMAT 5: 650 Q48 V31 GPA: 3
WE: Management Consulting (Consulting)
Re: How many two-digit whole numbers yield a remainder of 1 when divided  [#permalink]

### Show Tags

There is an easy way to solve this conceptually.

The only two digit numbers that will produce a remainder of 1 when divided by 10 are two-digit numbers with units digits of 1.
11, 21, 31, 41,51,61,71,81,91

You can quickly go through and determine that only 31,61,91 produce a remainder of 1 when divided by 6 and are two digit numbers.
_________________
Goal: Q49, V41

+1 Kudos if I have helped you Re: How many two-digit whole numbers yield a remainder of 1 when divided   [#permalink] 16 Oct 2019, 17:09
Display posts from previous: Sort by

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