How many pairs of natural numbers exist such that when the LCM of the numbers is added to their sum a total of 49 is obtained?
Find the number of ways to distribute three red balls, four blue balls and five green balls into four distinct boxes with no empty boxes.
Solve pls..
(32)^32^32 when divided by 9 will leave a remainder?
Let me know if there's a simple procedure.
There are m seats in the front row of a movie hall , n <m people enter and take seats randmly, probability that out of any two seats located symmetrically about the middle row , atleast one is empty,
What can be the maximum value of the curved surface area of a sphere inside a hemisphere of diameter 2 units?? Need the full approach..answer is 3.14
the remainder when 10^10+10^100+10^1000.........+10^10000000000 is divided by 7
Please give the working of both questions
For how many integral values of 'a' will the graphs of x^2+y^2=4x+8y+80 and x^2+y^2=10x+16y-89+a^2 meet each other at at least one point?
There are blue vessels with known volumes v1, v2..., vm, arranged in ascending order of volume, v1 > 0.5 litre, and vm < 1 litre. Each of these is full of water initially. The water from each of these is emptied into a minimum number of empty white vessels, each having volume 1 litre. The water from a blue vessel is not emptied into a white vessel unless the white vessel has enough empty volume to hold all the water of the blue vessel. The number of white vessels required to empty all the blue vessels according to the above rules was n."
Let the number of white vessels needed be n1 for the emptying process described above, if the volume of each white vessel is 2 litres. Among the following values, which is the least upper bound on n1?
A)
m/4
B)
smallest integer greater than or equal to n/2
C)
n
D)
greater integer less than or equal to n/2
PLEASE HELP, Ans is 30.
Amar, Bhavan and Chetan bought a circular pizza. They cut it into exactly 5 sectors, all of distinct sizes, and distributed these 5 parts among themselves such that each of them got at least 1 part but none of them got the parts of the pizza which were adjacent. In how many ways could they have shared the pizza?
Solve pls..
Please give solution
Please help
Boxes numbered 1, 2, 3, 4 and 5 are kept in a row, each of which are to be filled with one ball, either a red one or a blue one. No two adjacent boxes can be filled with blue balls. Then how many different arrangements are possible, given that all balls of the same colour are exactly identical in every respects?
Solve pls..
Smallest 2 digit no which will never be a perfect square in any base is???
How many ways a number can be written as a product of 2 natural no’s:
Case 1: Non square no:
Let’s take a simple no: 84.
Factors of 84 are:
1,2,3,4,6,7,12,14,21,28,42,84.
Total: 12 Let’s check if we miss any factor:
Total no of factors for 84: [(2^2)*3*7] = 3*2*2=12.
So, we captured all the factors of 84.
Now 84 can be expressed as a product of 2 natural no’s:
1*84, 2*42, 3*28, 4*21, 6*14, 7*12 ------- 6 Ways So if a no has n factors, it can be expressed as a product of 2 natural no’s as:
---- {(First factor from left)*(first factor from right)} ,
{(second factor from left)*(second factor from right)} ,
till………………………. {([n/2]nd factor from left) * ([n/2]nd factor from right)}.
So, if a no has ‘N’ factors, where N is even (for non square number), total no of ways it can be expressed as product of 2 natural no’s: N/2.
If you are clear till this point, you are thinking about what if a no has odd no of factors.
This leads to our case no 2.
Case 2: Square number: As many of you might be aware that Only a square number has odd no of factors. Or if a no has odd no of factors, then it has to be a square no.
lets take a simple no: 36
Factors of 36: 1,2,3,4,6,9,12,18,36.
Total no of factors: 9.
36 can be expressed as a product of 2 different natural no’s:
1*36, 2*18, 3*12, 4*9, 6*6. ------ 5 ways.
If you notice, middle no is expressed as a product to itself (6*6).
So, for a square no having no of factors as N, total no of ways are:
(N+1)/2 ------ If repetition of numbers is allowed.
(N-1)/2------- If both the no’s has to be unique.
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Pratik Shah(appeared for CAT 2016: DILR:99.86, Quant:95.9 percentile) Teaching Quant and DILR for CAT since April ,2017.
Contact on:
[email protected]
https://twitter.com/104868 Whatsapp no: +91 9834986902
How many ways a number can be expressed as a difference of two perfect squares of a natural no:
1. Odd number
2. Even no divided by 4
3. Even no not divided by 4.
Odd No:
take any odd no: for eg: 405
to express 405 as a difference of two perfect squares:
405= X2 –Y2 = (X-Y)(X+Y)
Replace X-Y=a and X+Y=b
so we need to find out in how many ways 405 can be expressed as a*b.
If you don’t know this concept, please read article:
https://www.pagalguy.com/discussions/hh-asd-fdf-fd-4771466512957440 No of Factors of 405 are: (34 * 51) = 5*2= 10
Factors(1,3,5,9,15,27,45,81,135,405).
So, no ways in which 405 can be expressed as product of 2 natural no’s= 10/2= 5.
(1*405,3*135,5*81,9*45,15*27)
So, no of ways in which 405 can be expressed as a difference of two perfect squares=5.
Let’s check: 405= 1*405.
X-Y=1
X+Y=405
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2X=406 ∴X=203 and Y=202.
∴ 405 can be expressed as a difference of two perfect squares:
2032 – 2022,
[(135+3)/2]2 - [(135-3)/2]2 = 692 -662
[(81+5)/2]2 - [(81-5)/2]2 = 432 -382.
[(45+9)/2]2 - [(45-9)/2]2 = 272 -182.
[(27+15)/2]2 - [(27-15)/2]2 = 212 -62.
From this we are clear that sum of 2 factors should be divided by 2. Hence both the factors should either be odd or even .As for odd no, all the factors are odd, an odd number can be expressed as a difference of two perfect squares= no of ways in which a number can be expressed as product of 2 natural no’s= N/2.
Where N=number of factors.
Case2:
Lets take even number now which is divisible by 4:
120= 1*120, 2*60, 3*40,4*30,5*24,6*20,8*15,10*12.
lets take a case 1*120.
1+120/2=121\2=60.5 and 120-1/2=59.5 .. Both of these are not natural no’s.
Hence we need to discard the cases where one factor is odd and one is even.
So only valid cases are: 2*60, 4*30 ,6*20, 10*12.
But, discard the cases where one factor is odd and one is even cannot be done manually each time.
Shortcut: as the no is even, at least one of the two factors will be even every time.
∴ we need to consider cases where both the factors are even . Hence express 120 as 2a’*2b’.
120=2a’ * 2b’
30=a’*b’
So no of ways to express 30 as product as two natural no’s = No of factors of 30/2 = 8/2 = 4 = no of ways in which 120 can be expressed as a difference of two perfect squares.
For a number which is divisible by 4:
a. Divide the no by 4.
b. Find the number of ways in which N/4 can be expressed as product of 2 natural nos.
Case 3:
Now consider a case where even number is not divisible by 4.
E.g.: take no as 50.
To find the answer, we need to express 50 as 2a’*2b’
Hence 50/4 = a’*b’. As 50 cannot be divided by 4, there won’t be any case in which 50 can be expressed as a product of 2 even natural nos.
So, an even no which is cannot be divided by 4: cannot be expressed as a difference of squares of 2 natural nos.
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Pratik Shah(appeared for CAT 2016: DILR:99.86, Quant:95.9 percentile) Teaching Quant and DILR for CAT since April ,2017.
Contact on:
[email protected]
https://twitter.com/104868
Whatsapp no: +91 9834986902