Last visit was: 06 Oct 2024, 18:53 It is currently 06 Oct 2024, 18:53
Close
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
Your Progress

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.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Show Tags
Hide Tags
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [53]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Most Helpful Reply
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [13]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
General Discussion
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [3]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
How to Solve: Divisibility Rules [#permalink]
1
Bookmarks
Expert Reply
Top Contributor
Attachment:
Divisibility Rules.pdf [148.5 KiB]
Downloaded 92 times
How to Solve: Divisibility Rules


Hi All,

I have posted a video on YouTube to discuss Divisibility Rules



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory
    What are Divisibility Rules and why are they useful?
    Divisibility Rule for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
    Similarity in Divisibility rule for 2, 4 and 8
    Similarity in Divisibility rule for 3 and 9
    Example Problems


What are Divisibility Rule and why are they useful?

    Find Factors: Divisibility Rules help us in quickly identifying if a number is a factor of another number or not.
    Save Time: If we use divisibility rules then we do not have to go for Long Division method to find out the factors of the number.

Divisibility Rule for divisibility by 2

There are multiple ways of checking if a number is divisible by 2 or not, 3 of them are listed below
    Number should be even
    Last digit of the number should be divisible by 2
    Units digit should be 0,2,4,6,8


Q1: Check if 360 is divisible by 2 or not.

Solution: There are multiple ways of checking it. 3 or them are given below:
    360 is even so 360 divisible by 3
    Last digit of 360 which is 0 is divisible by 2, so 360 divisible by 3
    Unit digit of 360 is 0, so 360 divisible by 3

Divisibility Rule for divisibility by 3

    Sum of all the digits of the number should be divisible by 3


Q2: Check if 360 is divisible by 3 or not.

Solution: Sum of all the digits of 360 = 3 + 6 + 0 =9
We know that 9 is divisible by 3 => 360 is divisible by 3

Divisibility Rule for divisibility by 4

    Number formed by last two digits should be divisible by 4


Q3: Check if 360 is divisible by 4 or not.

Solution: Number formed by last two digits of 360 is 60.
We know that 60 is divisible by 4 => 360 is divisible by 4

Divisibility Rule for divisibility by 5

    Number should end with 0 or 5


Q4: Check if 360 is divisible by 5 or not.

Solution: Since 360 ends with a 0 => 360 is divisible by 5

Divisibility Rule for divisibility by 6

    Number should be divisible by both 2 and 3


Q5: Check if 360 is divisible by 6 or not.

Solution: 360 is divisible by both 2 and 3 (Check above problems)
=> 360 is divisible by 6

Divisibility Rule for divisibility by 7

    Remove the last digit and double it and subtract it from the rest of the number.
    If the result is divisible by 7 then number is divisible by 7, else it is not


Q6: Check if 343 is divisible by 7 or not.

Solution: Yes 343 is divisible by 7. Check the video for detailed explanation.

Divisibility Rule for divisibility by 8

    Number formed by last three digits should be divisible by 8


Q7: Check if 1360 is divisible by 8 or not.

Solution: Number formed by last three digits of 1360 is 360.
We know that 360 is divisible by 8 => 1360 is divisible by 8

Divisibility Rule for divisibility by 9

    Sum of all the digits of the number should be divisible by 9


Q8: Check if 9360 is divisible by 9 or not.

Solution: Sum of all the digits of 9360 = 9 + 3 + 6 + 0 = 18
We know that 18 is divisible by 9 => 9360 is divisible by 9

Divisibility Rule for divisibility by 10

    Number should end with 0


Q9: Check if 360 is divisible by 10 or not.

Solution: Since 360 ends with a 0 => 360 is divisible by 10

Divisibility Rule for divisibility by 11

    If the difference of the sum of odd place digits and the sum of even place digits of the number is divisible by 11, then the number is divisible by 11, else it is not


Q10: Check if 1320 is divisible by 11 or not.

Solution: In 1320
Sum of odd places = 1 + 2 = 3
Sum of even places = 3 + 0 = 3

Sum of odd places - sum of even places = 3-3 =0
And 0 is divisible by all the numbers
=> 1320 is divisible by 11

Divisibility Rule for divisibility by 12

    Number should be divisible by both 3 and 4


Q11: Check if 360 is divisible by 12 or not.

Solution: 360 is divisible by both 3 and 4 (Check above problems)
=> 360 is divisible by 12


Similarity in Divisibility rule for 2, 4 and 8

    2 can be written as \(2^1\) -> Rule -> Number formed by last 1 digit(s) should be divisible by 2
    4 can be written as \(2^2\) -> Rule -> Number formed by last 2 digit(s) should be divisible by 4
    8 can be written as \(2^3\) -> Rule -> Number formed by last 3 digit(s) should be divisible by 8


Similarity in Divisibility rule for 3 and 9

    Divisibility Rule for 3: Sum of all the digits of the number should be divisible by 3
    Divisibility Rule for 9: Sum of all the digits of the number should be divisible by 9


Sample Problems

Q12: 336 is divisible by which of the following (multiple options correct)?
A. 2
B. 3
C. 6
D. 7
E. 9

Answer: A,B, C, D. [ Check Video for Explanation ]

Q13: 3773 is divisible by which of the following (multiple options correct)?
A. 2
B. 3
C. 7
D. 9
E. 11

Answer: C, E [ Check Video for Explanation ]

Hope it helps!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
How To Solve: Factors and Multiples [#permalink]
1
Kudos
Expert Reply
Top Contributor
Show SpoilerDOWNLOAD PDF: How To Solve: Factors and Multiples
Attachment:
Factors and Multiples.pdf [150.24 KiB]
Downloaded 40 times

How To Solve: Factors and Multiples


Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Hi All,

I have recently uploaded a video on YouTube to discuss Factors and Multiples in Detail:




Following is covered in the video

    ¤ What are Multiples?
    ¤ What are Factors?
    ¤ How to find the number whose two factors are given.
    ¤ Relationship between divisor, factor and multiple

What are Multiples?

We say that “a” is a multiple of “b” when we can express “a” as a product of “b” and any integer

    a = kb , where k is an integer

Example 1: 6 is a multiple of 2 because we can express 6 as a product of 2 and an integer (3). (6 = 2*3)

Example 2: “a number(n) is a multiple of 3”.
=> n = 3*k (where k is an integer)


What are Factors?

We say that a number “a” is a factor of a number “b” when we can write “b” as a multiple of “a”

    b = ka , where k is an integer

Example 1: 2 is a factor of 6 because we can write 6 as a multiple of 2 i.e. 6 = 2*3

Example 2: “a number(n) has 3 as one of its factor's”.
So, n is a multiple of 3.
=> n = 3*k (where k is an integer)

How to find the number whose two factors are given.

If a number has 2 and 3 as its factors then the number will be a multiple of LCM of these two factors

    n = LCM (2,3) k = 6k


Relationship between divisor, factor and multiple

If n is divisible by 3 then

    ¤ 3 is a factor of n
    ¤ n is a multiple of 3
    ¤ n can be written as, n = 3k, where k is an integer

Hope it helps!
Good Luck!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
How To Solve: Number and Sum of Factors of a number [#permalink]
1
Kudos
Expert Reply
Top Contributor
Show SpoilerDOWNLOAD PDF: How To Solve: Number and Sum of Factors of a number
Attachment:
Number and Sum of Factors.pdf [163.49 KiB]
Downloaded 81 times

How To Solve: Number and Sum of Factors of a number


Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Hi All,

I have recently uploaded a video on YouTube to discuss Number and Sum of Factors of a number in Detail:




Following is covered in the video

    ¤ How to find Number of Factors of a number.
    ¤ Solved Problems: Number of Factors
    ¤ How to find Sum of Factors of a number.
    ¤ Solved Problems: Sum of Factors

How to find Number of Factors of a number.

To find number of factors of a number we need to write the number as product of power of prime number and add one to the powers and multiply the powers.

Example : Find the number of factors of 72.

72 = \(2^3 * 3^2\)
=> Number of factors of 72 = (3+1) * (2+1) = 4 * 3 = 12

We can list down the factors as
1 * 72
2 * 36
3 * 24
4 * 18
6 * 12
8 * 9


Solved Problems: Number of Factors

Q1 : Find number of factors of 100 and list the factors?

Sol: 100 = \(2^2 * 5^2\)
=> Number of factors of 100 = (2+1) * (2+1) = 3 * 3 = 9

We can list down the factors as
1 * 100
2 * 50
4 * 25
5 * 20
10 * 10

Q2 : Find number of factors of 8?

Sol: 8 = \(2^3\)
=> Number of factors of 8 = (3+1) = 4

We can list down the factors as 1, 2, 4, 8


How to find Sum of Factors of a number

Sum of factors of an integer n, where n = \(p^a * q^b * r^c\) (p, q, r being prime number) is given by

\(\frac{(p^{a+1} – 1) * (q^{b+1} – 1) * (r^{c+1} – 1)}{ (p-1) * (q-1) * (r-1)}\)

Solved Problems: Sum of Factors

Q1 : Find the sum of Factors of 60.

Sol: 60 = \(2^2 * 3^1 * 5^1\)
Sum of factors of 60 = \(\frac{(2^{2+1} – 1) * (3^{1+1} – 1) * (5^{1+1} – 1)}{ (2-1) * (3-1) * (5-1)}\) = \(\frac{7 * 8 * 24}{ 1 * 2 * 4}\) = 168

Hope it helps!
Good Luck!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1954
Own Kudos [?]: 2217 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Factors and Multiples Question Set (Easy-Medium) [#permalink]
1
Kudos
Expert Reply
Top Contributor
Show SpoilerDOWNLOAD PDF: Factors and Multiples Question Set (Easy-Medium)
Attachment:
Factors and Multiples Solved Problems.pdf [171.87 KiB]
Downloaded 65 times

Factors and Multiples Question Set (Easy-Medium)


Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Hi All,

I have recently uploaded a video on YouTube to discuss Factors and Multiples Question Set (Easy-Medium) in Detail:




Seven solved problems are covered in the video.

Q1. If 26 is a factor of n, then is 6 a factor of n?

Sol: 26 is a factor of n => n is a multiple of 26
=> n = 26k where k is an integer

For 6 to be a factor of n, 26k should be divisible by 6
We need a 2 and a 3 for 6, but we have only 2 in 26k (=2 * 13k)

So, 26k will be divisible by 6 only when k is divisible by 3, which might not be true always
=> We CANNOT say for sure if 6 is a factor of n.

Q2. If 90 is a factor of n, then is 20 a factor of n?

Sol: 90 is a factor of n => n is a multiple of 90
=> n = 90k where k is an integer

For 20 to be a factor of n, 90k should be divisible by 6
We need a 2 and a 10 for 20, but we have only 10 in 90k (=10 * 9k)

So, 90k will be divisible by 20 only when k is divisible by 2, which might not be true always
=> We CANNOT say for sure if 20 is a factor of n.

Q3. If 24 is a factor of n, then is n a multiple of 8?

Sol: 24 is a factor of n => n is a multiple of 24
=> n = 24k where k is an integer

n = 24k = 8 * 3k
=> Clearly, n is a multiple of 8 => TRUE.

Q4. If 4 and 6 are factors of x, then which of the following will be a factor of x for sure
A. 8
B. 12
C. 24
D. 36
E. 48


Sol: 4 and 6 are factors of n => n is a multiple of LCM(4,6)
=> n = LCM(4,6) * k where k is an integer
= 12k
=> n is a multiple of 12 for sure

=> So, Answer will be D.

Q5. Is 192 divisible by x?
A. x is divisible by 12
B. x is divisible by 16


Sol:
STAT A: x is divisible by 12
=> x will be a multiple of 12.
=> x can be 12, 24, 192 or even 192 *10.
When x is 12, 24, 192 then 192/x is an integer but when x is 1920 then 192/x is not an integer.
=> NOT SUFFICIENT

STAT B:x is divisible by 16.
=> x will be a multiple of 16.
=> x can be 16, 32, 192 or even 192 *10.
When x is 16, 32, 192 then 192/x is an integer but when x is 1920 then 192/x is not an integer.
=> NOT SUFFICIENT

Taking both the statements together we get

Combining both also gives us that x will be a multiple of LCM(12,16)= 48k.
=> x can be 48,..192, 1920.
When x is 48, 96, 192 then 192/x is an integer but when x is 1920 then 192/x is not an integer.
=> NOT SUFFICIENT

=> So, Answer will be E.

Q6. If 12 is a factor of x and 16 is a factor of y, then
6.1 Is 48 a factor of xy?
6.2 Is 96 a factor of xy?
6.3 Is 192 a factor of xy?


Sol: 12 is a factor of x
=> x is a multiple of 12 => x = 12a
16 is a factor of y
=> y is a multiple of 16 => y = 16b
xy = 12a * 16b = 192 ab

So, xy is a multiple of
48 because xy = 48*4*ab
96 because xy = 96*2 *ab
192 because xy= 192*1*ab

So, Answer will be TRUE for all three problems

Q7. In a competition, gold medal gives 7 points, Silver gives 5 points and bronze gives 3 points. Roger participated in the competition in various events and the product of all the points earned by him is 33075. Then find the number of gold, silver and bronze medal earned by Roger

Sol: Bronze = x, Silver = y, Gold = z
Product of all the points = 33075
=> 33075 = \(3^x * 5^y * 7^z\)
33075 = \(3^3 * 5^2 * 7^2\)
\(3^3 * 5^2 * 7^2\) = \(3^x * 5^y * 7^z\)

So, Answer will Roger won: Gold – 2, Silver – 2, Bronze - 3


Watch the following video to learn the Basics of Factors and Multiples




Hope it helps!
Good Luck!
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 35130
Own Kudos [?]: 891 [0]
Given Kudos: 0
Send PM
Re: How to Solve: Remainders (Basics and Advanced) [#permalink]
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.
GMAT Club Bot
Re: How to Solve: Remainders (Basics and Advanced) [#permalink]
Moderator:
Math Expert
95949 posts