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Types of Numbers
by Total Gadha - Friday, 9 February 2007, 07:06 PM
 

Natural Numbers:

The group of numbers starting from 1 and including 1, 2, 3, 4, 5, and so on. Zero, negative numbers, and decimals are not included this group.

EXAMPLE

1. If n is an odd natural number, what is the highest number that always divides n(n2 – 1)?

Answer: n∙(n2 – 1) = (n – 1)∙n∙(n + 1), which is a product of three consecutive numbers. Since n is odd, the numbers (n – 1) and (n + 1) are both even. One of these numbers will be a multiple of 2 and the other a multiple of 4 as they are two consecutive even numbers. Hence, their product is a multiple of 8. Since one out of every three consecutive numbers is a multiple of 3, one of the three numbers will be a multiple of three. Hence, the product of three numbers will be a multiple of 8 ´ 3 = 24.

Hence, the highest number that always divides n∙(n2 – 1) is 24.

2. For every natural number n, the highest number that n∙(n2 – 1)∙(5n + 2) is always divisible by is

(a) 6 (b) 24 (c) 36 (d) 48

Answer:

Case 1: If n is odd, n∙(n2 – 1) is divisible by 24 as proved in the earlier question.

Case 2: If n is even, both (n – 1) and (n + 1) are odd. Since product of three consecutive natural numbers is always a multiple of 3 and n is even, the product n∙(n2 – 1) is divisible by 6. Since n is even 5n is even. If n is a multiple of 2, 5n is a multiple of 2 and hence 5n + 2 is a multiple of 4. If n is a multiple of 4, 5n + 2 is a multiple of 2. Hence, the product n∙(5n + 2) is a multiple of 8.

Hence, the product n∙(n2 – 1)∙(5n + 2) is a multiple of 24.

Hence, [b]

Rule: The product of n consecutive natural numbers is divisible by n!, where n! = 1 Ã— 2 Ã— 3 Ã— 4 Ã— 5…. Ã— n

EXAMPLE

3. Prove that (2n)! is divisible by (n!)2.

Answer: (2n)! = 1·2·3·4· … ·(n – 1)·n·(n + 1)· …·2n

= (n)!·(n + 1)·(n + 2)· …·2n.

Since (n + 1)·(n + 2)· …·2n is a product of n consecutive numbers, it is divisible by n!. Hence, the product (n)!·(n + 1)·(n + 2)· …·2n is divisible by n!·n! = (n!)2.

Whole Numbers:

All Natural Numbers plus the number 0 are called as Whole Numbers.

Integers:

All Whole Numbers and their negatives are included in this group.

Rational Numbers:

Any number that can be expressed as a ratio of two integers is called a rational number.

This group contains decimal that either do not exist (as in 6 which is 6/1), or terminate (as in 3.4 which is 34/10), or repeat with a pattern (as in 2.333... which is 7/3).

Irrational Numbers:

Any number that can not be expressed as the ratio of two integers is called an irrational number (imaginary or complex numbers are not included in irrational numbers).

These numbers have decimals that never terminate and never repeat with a pattern.

Examples include pi, e, and √2. 2 + √3, 5 - √2 etc. are also irrational quantities called Surds.

EXAMPLE

example 4

Example 5

Real Numbers:

This group is made up of all the Rational and Irrational Numbers. The ordinary number line encountered when studying algebra holds real numbers.

Imaginary Numbers:

These numbers are formed by the imaginary number i (i = √-1). Any real number times i is an imaginary number.

Examples include i, 3i, −9.3i, and (pi)i. Now i2 = −1, i3 = i2 Ã— i = −i, i4 = 1.

EXAMPLE

Example

Complex Numbers:

A Complex Numbers is a combination of a real number and an imaginary number in the form a + bi. a is called the real part and b is called the imaginary part.

Examples include 3 + 6i, 8 + (−5)i, (often written as 8 - 5i).

image

Prime Numbers:

All the numbers that have only two divisors, 1 and the number itself, are called prime numbers. Hence, a prime number can only be written as the product of 1 and itself. The numbers 2, 3, 5, 7, 11…37, etc. are prime numbers.

Note: 1 is not a prime number.

EXAMPLE

  1. If x2 – y2 = 101, find the value of x2 + y2, given that x and y are natural numbers.

Answer: x2 – y2 = (x + y)(x – y) = 101. But 101 is a prime number and cannot be written as product of two numbers unless one of the numbers is 1 and the other is 101 itself.

Hence, x + y = 101 and x – y = 1. -> x = 51, y = 50.

-> x2 + y2 = 512 + 502 = 5101.

  1. What numbers have exactly three divisors?

Answer: The squares of prime numbers have exactly three divisors, i.e. 1, the prime number, and the square itself.

note

Odd and Even Numbers:

All the numbers divisible by 2 are called even numbers whereas all the numbers not divisible by 2 are called odd numbers. 2, 4, 6, 8… etc. are even numbers and 1, 3, 5, 7.. etc. are odd numbers.

note

Re: Types of Numbers
by anil ahirwal - Tuesday, 12 June 2007, 01:08 PM
  Hmmmmmmmm, Nice
Re: Types of Numbers
by bhavin gohel - Tuesday, 24 July 2007, 01:33 PM
  Hey just joined your forum....found amazing information from it.....looks very useful..... thanks for such wonderful threads and keep them coming
Re: Types of Numbers
by Parag Lavane - Thursday, 30 August 2007, 07:17 PM
  Very useful thank you
Re: Types of Numbers
by ashok goel - Friday, 31 August 2007, 03:53 AM
  hi
thanks for such  informative  articles
can i get articles before july 07 also
solve
by shyam Sundar - Sunday, 9 September 2007, 04:25 PM
 

if 21 base n * 36 base n = 776   and 12  base n *  63 base n  = x base n  solve for x

 

510

540

776

756

 

 

Re: solve
by shyam Sundar - Sunday, 9 September 2007, 04:26 PM
 

a no wen divided by a certain divisor  leaves remauinder 8 . wen same numbr is multipld by 12 and divided by same diviosr remainder is 12 , how many such divisors are posibe l

 

 

explain with steps

 

1

2

3

4

 

 

Re: solve
by shyam Sundar - Sunday, 9 September 2007, 04:30 PM
 

100 players from 1 to 100 n 100 baskets frm 1 o 100

1st player puts 1 ball i evry bask strtin frm d first bask , second puts 2 in each an so on  strtin frm the second basket ( in baskets 2 , 4 , 6 ..)  

   third palyer puts 3 in baskets in 3 , 6 , 9 .. .... 

 

proces continues til 100th player puts 100 bals in 100th baskeet

 

wich basket wil hv max no of balls

 

96

98

100

none f these

 

how many baskts hv twice the no of balls as the basket no

 

8

 

6

 

4

 

2

 

pls gv steps

 

 

Re: solve
by shyam Sundar - Sunday, 9 September 2007, 04:32 PM
 

6 ppl share a piza by cutin it out nto 6 equal sectors . if 3 of dem cutout and eat only the largest posibel circle from their respective slices and leavve d rest while otheres eat the  whole slice wats d percentage of piza wasted

 

11

15

 

17

 

22

steps pls

 

Re: solve
by rohit avasthi - Monday, 10 September 2007, 04:43 PM
 

ans 17.

its actually coming out as 16.667.

let 'L' be the pizza radius,r be the inner circle radius then,

(L-r)sin30'=r.....by geometry

thus,r = L/3...hence get the answer

for shyam sundar's
by rohit avasthi - Monday, 10 September 2007, 05:03 PM
 

ans 96.

answer clearly lies in 90-100 bracket,since answer is the sum of factors of the basket no.

96 can be expressed in terms of products of prime nos. in a way that it gives out max. no of factors.

96=2^5 * 3^1......hence no. of factors is 12,which is maximum for the 90 to 100 nos.

 

My process to solve the 2nd part of question may not seem convincing to u.

rohit

to shyam sundar's(no. of divisors)
by rohit avasthi - Monday, 10 September 2007, 05:36 PM
 

answer seems to be 5 (:-?)

the conditions given can be expressed as : (for divisor = X )

condn1: (N-8) / X =M...where m and n are integers.

condn2.: (12N-12) / X = L  ...where L is an integer.

from 1: (12N - 96) / X = 12M = A...some integer constant.

ie. 12 X = A X + 96 = LX + 12

OR, (L-A)X = 84....

NOW THAT WE KNOW , X>12 SINCE REMAINDER CAN HAVE THE VALUE OF 12

HENCE THE NO. OF WAYS OF DIVIDING THE NUMBER 84 AS 2 PRODUCTS ARE:

14 X 6

21 X 4

28 X 3

42 X 2

84 X 1

 

HENCE IT CAN HAVE THE VALUES OF 14,21,28,42 AND 84

ROHIT

Re: Types of Numbers
by shahab siddiqui - Monday, 10 September 2007, 05:57 PM
  very useful waiting from long time....
Re: solve to shyam sundar's (21 base n * 36 base n = 776)
by rohit avasthi - Monday, 10 September 2007, 06:05 PM
 

answer is 776,

follow TG's base system paper...u will find this question  simple...

On  reasoning u will find out that the base value is n = 8

for which answer is 776.

Re: solve
by shyam Sundar - Monday, 10 September 2007, 06:49 PM
 

rohit

 

could u eelaborate a lil more on dat by providin a jpg daigram or sumtin lik dat

hw did u gt (L-r)sin30'=r , wat d u mean by inner circle redius ??

 

obviusly the length of the small sectors is goin 2 b the radius  of the pizza !

pls elaborate 

and how did u suddenly reach 16.67??

 

 

Re: solve
by Software Engineer - Monday, 10 September 2007, 07:17 PM
  if 21 base n * 36 base n = 776 base n   and 12  base n *  63 base n  = x base n  solve for x
510 540 776 756

(2n + 1) (3n + 6) = 7n^2 + 7n + 6
n=8
so, 12 base 8 * 63 base 8 = 776 base 8
ans. 8


a no wen divided by a certain divisor  leaves remauinder 8 . wen same numbr is multipld by 12 and divided by same diviosr

remainder is 12 , how many such divisors are posibe 1 2 3 4

n = x * d + 8 and 12n = y * d + 12
multiply [1] by 12 , equatw with [2]
y * d + 12  = 12 * x * d + 12 * 8
d ( y - x) = 96 - 12 = 84
d (m) = 84
divisiors of 84 : 1,2,3,4,6,7,12,14,21,28,42,84
as could be 8 or 12, so d must be greater than 12
divisors of 84 greater than 12 (excluding itself) : 14, 21,28,42 ( d can take any of this value)
so ans. 4


100 players from 1 to 100 n 100 baskets frm 1 o 100
1st player puts 1 ball i evry bask strtin frm d first bask , second puts 2 in each an so on  strtin frm the second basket ( in

baskets 2 , 4 , 6 ..)  
   third palyer puts 3 in baskets in 3 , 6 , 9 .. ....
proces continues til 100th player puts 100 bals in 100th baskeet
wich basket wil hv max no of balls
96  98  100  none f thes

a number which has max. no. of divisors has max. balls
from the optionssuch no. is 96

how many baskts hv twice the no of balls as the basket no
8  6 4 2

could not get the que. could any1 tell what is it saying?
Re: software engineer(21 base n * 36 base n = 776 base n)
by rohit avasthi - Wednesday, 12 September 2007, 10:09 AM
 

u hav solved it right but they havnt asked for the value of n....they asked for value of "x".....which as i mentioned earlier is 776.

continue this thread...

cya,,,,rohit

Re: software engineer(21 base n * 36 base n = 776 base n)
by Software Engineer - Wednesday, 12 September 2007, 11:14 AM
  oooooooooopsmixed
can u ans my que.
Re: solve
by Chims... Naik - Saturday, 19 April 2008, 09:05 PM
 

100 players from 1 to 100 n 100 baskets frm 1 o 100

1st player puts 1 ball i evry bask strtin frm d first bask , second puts 2 in each an so on  strtin frm the second basket ( in baskets 2 , 4 , 6 ..)  

 third palyer puts 3 in baskets in 3 , 6 , 9 .. .... 

 proces continues til 100th player puts 100 bals in 100th baskeet

Q: how many baskts hv twice the no of balls as the basket no

 8

 6

 4

 2

 Solution:

For the balls in the basket to be twice the number displayed on the basket, the sum of the divisors of the number displayed on the basket has to be equal to the number itself. Thus the problem reduces to finding the perfect nos. between 1 to 100. There are only two perfect nos.(6 and 28) between 1 and 100. So 2 such baskets are possible.

1) Divisors of 6 -> 1, 2, 3, 6 (sum=1+2+3+6=12=2*6)
2) Divisors of 28 -> 1, 2, 4, 7, 14, 28 (sum=1+2+4+7+14+28=56=2*28)
Re: Types of Numbers
by Abhimanyu Singh Sisodia - Saturday, 26 April 2008, 04:02 PM
 

it was very nice

thanks

Re: Types of Numbers
by love kumar - Monday, 19 May 2008, 04:50 PM
 

can anyone solve dis prob?

Let N= 2^23 * 3^21. How many positive divisors of N^2 are less than N but don`t divide N?

Re: solve
by Karan Menda - Tuesday, 27 May 2008, 11:54 AM
  Good one...
Re: solve
by Aditya Zutshi - Tuesday, 27 May 2008, 08:00 PM
 

Hi TG,

In Q2, Case 2 you've mentioned that "5n is a multiple of 2 and hence 5n + 2 is a multiple of 4". I had a confusion in this. If we put n as 4,8 etc then (5n+2) isn't divisible by 4. Pls reply...

Re: solve
by Total Gadha - Tuesday, 27 May 2008, 11:19 PM
  Hi Aditya,

These are the two separate cases:
  • 5n is a multiple of 2
  • 5n is a multiple of 4
One golden rule that you should remember is that in any two consecutive even numbers one is a multiple of 4 and the other is a multiple of 2.

So here's the conclusion:
  • 5n is a multiple of 2 - 5n + 2 is a multiple of 4
  • 5n is a multiple of 4 - 5n + 2 is a multiple of 2

Total Gadha
Re: solve
by Aditya Zutshi - Saturday, 31 May 2008, 05:02 PM
 

Hi TG,

Thanks a ton !!! I got the concept !!!

Hats off to u smile Don't know wat I'd have done without u guys !!! I'm addicted to TG.com now !!! Can u beat tat!!! smile Thanks.

Re: Types of Numbers
by Deepak tewari - Sunday, 1 June 2008, 11:00 AM
  gud one....
Re: Types of Numbers
by Willy Will - Saturday, 7 June 2008, 10:29 PM
  I hav come across various sites/forums for CAT prep but this outperforms all of them.
TG rocks ....
Types of Numbers
by rajesh deo - Monday, 9 June 2008, 10:19 AM
 

hi..........

firstly i thanks for giving such an important concept on number system ,which i don't know previously

Types of Numbers
by S A R SHAH - Friday, 13 June 2008, 08:30 PM
 

A nice one 

 

Re: solve
by Jasmeet Singh suri - Monday, 16 June 2008, 03:44 PM
  the answer is 776  as 21*36=12*63
Re: Types of Numbers
by Vamsi Krishna Kancherla - Monday, 16 June 2008, 07:20 PM
  cool one with complete picture on Types of Numberssmile
Re: Types of Numbers
by Chitrang Dalal - Tuesday, 17 June 2008, 12:00 AM
 

can anyone solve dis prob? Let N= 2^23 * 3^21. How many positive divisors of N^2 are less than N but don`t divide N?

SOLUTION:

factors of N= (23+1)(21+1)=24*22=528

N^2 = 2^46 * 3^42

factors of N^2= 47*43=2021....factors of N^2 includes N.....excluding N factors of N^2= 2020

factors of N^2 less than N= 1010

factors of N^2 wich are less than N and do not divide N =1010 -527(Since N is to be excluded)

ANS: 483............

cheers!!!!

Re: Types of Numbers
by rakesh ojha - Tuesday, 17 June 2008, 04:18 PM
 

thnx alot for providing such rare information....

Re: Types of Numbers
by manoj 1123 - Monday, 7 July 2008, 04:25 PM
  Hi TG sir,
Thanks a lot for such useful information.
Can we proceed the exp - 5 in a faster way like,
(p+q)**2 - pq = 30**2 -1 = 899.
Thanks.
Re: Types of Numbers
by Amey Panke - Wednesday, 16 July 2008, 01:43 AM
 
Sir can u plzz solve this question for me......

  IF S=[(104+324)*(224+324)*(344+324)*(464+324)*(584+324)]/[(44+324)*(164+324)  *(284+324)*(404+324)*(524+324)].

Find S


thank you
Re: Types of Numbers
by praveen gattupalli - Saturday, 19 July 2008, 10:15 PM
  mind blowing sirjiiiiiiiiiii
Re: Types of Numbers
by Subhendu ... - Monday, 29 September 2008, 09:34 AM
  Hi Amey,
You might have found the soln.But this is the way how i solved this prob.

Take last two digits if each term..

S = (24 * 80 * 60 * 80 * 20) / (80 * 60 * 80 * 24 * 40) = last digits (2 / 4) = last digit should be 3 or 8.

correct ans 373…

Re: solve
by manish sharma - Saturday, 4 October 2008, 01:01 PM
  ans is 776
Re: Types of Numbers
by sachin yelapure - Monday, 24 August 2009, 05:23 PM
 

Could you please explain the logic behind the following line from your solution....a detailed explanation, if given, would be admired...

"factors of N^2 less than N= 1010"

Re: Types of Numbers
by Ankur Agarwal - Sunday, 3 October 2010, 12:46 AM
 

Hi TG,

I am a joined your forum few days back and really become ur big fan in very short time. Hats-off for your contribution to such a competitive world where quality was gradually degrarded but you have prevented it from sinking.

I need a little favour from you, I read this (Types of Numbers) article but barely I am getting the equations as I am encountering some difficulties in alphabets like ^n~"` etc etc...I think you can also see such problems in this article....

Please Please Please send me the article in attachment or other means so that I can follow the it properly.

Re: solve
by saanthwana T - Saturday, 7 May 2011, 10:19 PM
  Could you please let me know what a perfect number means?



Re: solve
by srinivas hosur - Sunday, 8 May 2011, 12:01 AM
  hi saanthwana,
if N is a perfect no. then....
sum of all divisors of N including 1 and excluding the no. itself is equal to N.
such no. take the form (2^(p-1))*((2^p)-1),where p is a prime no...
it can also be said that perfect no. is always even.
eg:6,1+2+3=6
Re: Types of Numbers
by srinivas hosur - Sunday, 8 May 2011, 12:41 AM
  hi kumar,
Let N= 2^23 * 3^21. How many positive divisors of N^2 are less than N but don`t divide N?
the answer for this is 1 and that no. is 2^46.
u can understand this by yourself by taking small values of N.
this is contradicting other's answers but u can verify that,the above soln. is the correct one.
Re: Types of Numbers
by srinivas hosur - Sunday, 8 May 2011, 07:12 PM
  hi kumar,
Let N= 2^23 * 3^21. How many positive divisors of N^2 are less than N but don`t divide N?
i am sorry for my ans.its totally absurd
Re: Types of Numbers
by destiny unruled - Monday, 9 May 2011, 11:20 PM
  @ Srinivas

Total number of factors of N = 24*22 = 528
Total number of factors of N² = 47*43 = 2021

Now, we know that there will be equal number of factors of N² greater than N and less than N.

=> No of factors of N² less than N = (2021 - 1)/2 = 1010

Now, these 1010 factors also includes 528 factors (factors of N)

=> No of factors of N² that are less than N and are not factors of N will be (1010 - 528) = 482
Re: Types of Numbers
by TG Team - Tuesday, 10 May 2011, 03:32 PM
 

Hi destiny unruled smile

Slight error. Number of factors of N less than N = 528 - 1 = 527.

So the required answer will be 1010 - 527 = 483. smile

Kamal Lohia

Re: Types of Numbers
by destiny unruled - Tuesday, 10 May 2011, 05:59 PM
  Yeah made a mistake. Thank you Sir smile
Re: Types of Numbers
by Rajasekaran Rajaram - Wednesday, 11 May 2011, 09:49 PM
  Kamal Lohia,

Sorry to interrupt. I am not sure where to post my clarifications. Please forgive me if i break the link and mess up sad

Q: Find the last two digits of 50! and 90!. Please help me with the reasoning involved.
Re: Types of Numbers
by nikita dhanuka - Thursday, 12 May 2011, 12:07 AM
  the last two digits are 00. smile

50!= 50*49*48*47...3*2*1
it is a huge number which ends in multiple zeros. similar case for 90!.
a zero results from a combination of 2 & 5. in a factorial obviously 2's are in greater abundance. hence, the number of zeros is limited by the number of 5's.
50! ends in 12 zeros.
90! ends in 21 zeros.

-nikita
Re: Types of Numbers
by Ganesh B - Thursday, 12 May 2011, 07:14 AM
  @Rajsekaran:

In the Home page of TotalGadha.com, go through KL Math Corner, these concepts have been extensively dealth with.......
Re: Types of Numbers
by Rajasekaran Rajaram - Thursday, 12 May 2011, 03:05 PM
  Forgive me for the mistake.

The question is to find the last two non zero digits of 50! and 90!
Re: Types of Numbers
by nikita dhanuka - Wednesday, 25 May 2011, 05:17 PM
  go through the theory that Kamal has discussed on the following page:

http://totalgadha.com/mod/forum/discuss.php?d=8392

its a great article. i am sure you will be able to solve the sums yourself, after reading it smile
Re: Types of Numbers
by ashu yadav - Tuesday, 7 June 2011, 12:59 PM
  what is this meaning of …·2n this in article i didn't understand sad