Need help with these QT questions from Arun Sharma's book

indian_girl · 07:34 PM

Hello,

I am unable to solve certain problems and I m posting few of them in this thread,

I request anyone who can solve them out to provide with a step by step explanation for the same such that it can be helpful not only to me but to any one who reads them out to solve them as well as similar kind of problems...

The following problems are from the chapter named "Number Systems"

Set 1:
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Q1. A number 15B is divisible by 6. which of these will be true about the positive integer B?
a. B will be even
b. B will be odd
c. B will be divisible by 6
d. Both a & c

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Q2. Find the units digit of the expression 55^725+ 73^5810 + 22^853 .

a. 4 b. 0 c.6 d.5

Note: 55^725 implies 55 to the power of 725 and not 55 multiplied by 725. same applies to all.

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Q3. Find the last digit in the expression (36472)^123! x (34767)^76!

~ This is an example problem so no options are available for this one ~

Note: 36472 ^123! implies 36472 to the power of 123! and same applies to all

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Q4. There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls . find the total number of sections formed?

a.24 b. 32 c. 16 d. none

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Q5. For what maximum value of n will the expression 10200!/ (504^n) be an integer?

~ No options available for this problem as its an example problem ~

Note: 504^n implies 504 to the power of n

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Thank you.

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Here , is the file that contains the solutions for this Set,If one has any suggestions or any possible corrections to be made in this solution set , pls feel free to post......thank you.

TheCat

Quote:

Originally Posted by indian_girl

Hello,

I am unable to solve certain problems and I m posting few of them in this thread,

I request anyone who can solve them out to provide with a step by step explanation for the same such that it can be helpful not only to me but to any one who reads them out to solve them as well as similar kind of problems...

The following problems are from the chapter named "Number Systems"
-------------------------------------------------------------------

Q1. A number 15B is divisible by 6. which of these will be true about the positive integer B? Ans: c. B will be divisible by 6

a. B will be even
b. B will be odd
c. B will be divisible by 6
d. Both a & c

-------------------------------------------------------------------

Q2. Find the units digit of the expression 55^725+ 73^5810 + 22^853 . Ans:c. 6

a. 4 b. 0 c.6 d.5

Note: 55^725 implies 55 to the power of 725 and not 55 multiplied by 725. same applies to all.

-------------------------------------------------------------------

Q3. Find the last digit in the expression (36472)^123! x (34767)^76! Ans: 6

~ This is an example problem so no options are available for this one ~

Note: 36472 ^123! implies 36472 to the power of 123! and same applies to all

-------------------------------------------------------------------

Q4. There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls . find the total number of sections formed? Ans: c. 16

a.24 b. 32 c. 16 d. none

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Q5. For what maximum value of n will the expression 10200!/ (504^n) be an integer? Ans:1698

~ No options available for this problem as its an example problem ~

Note: 504^n implies 504 to the power of n

-------------------------------------------------------------------

Thank you.

don't mind..please don't provide answers beforehand....ok now your solutions..

1. what do you mean by 15B here is it 15*B yaa number of the form 15B..
see if its 15*B then the answer you provide is wrong..as B only even will satisfy 15B divisible by 6 ex.take B=2..then 15B = 30..divisible by 6

other answers in subsequent post..

TheCat

Q2...unit digit of 55^725+ 73^5810 + 22^853

see unit digit of 55^725=5
unit digit of 73^5810 is equivalient to unit digit of 3^5810 = 9
unit gigit of 22^853 is equivalient to unit digit of 2^853=2

hence unit digit of the expression=5+9+2=16 ..hence 6

Q3.Find the last digit in the expression (36472)^123! x (34767)^76!

ans: unit digit of (36472)^123! is equivalent to unit digit of 2^123!
since 123! is divisible by 4 hence unit digit of 2^123!=6
following above step unit digit of (34767)^76! = 1

hence unit digit of expression = 6*1=6 ans..

Hope above steps are clear..if in doubt pm me...

TheCat

Q4.There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls . find the total number of sections formed? Ans: c. 16
Ans:
find the hcf of 576 and 448 = 64..
no. of rooms required = 576/64 + 448/64 = 9+7 = 16 rooms..

Q5.For what maximum value of n will the expression 10200!/ (504^n) be an integer?

Ans: 504 = 2^3*3^2*7

now find out the highest power of 2^3, 3^2 and 7 that will divide 10200! and choose the least one as answer....and you will find the least one for 7 and its equal to 1698..hrnce answer...

hope above steps are clear..

PS: easy question by last two year cat standard...

indian_girl

Hi Cat,

nah! nothing to my mind there, infact I think you are right I will just edit the post and remove the answers ,

and thanx for providing solutions they are very helpful but would need some more explanation on that , sorry for my poor maths

I have one doubt here :

Quote:

Originally Posted by TheCat

Q2...unit digit of 55^725+ 73^5810 + 22^853

unit digit of 73^5810 is equivalient to unit digit of 3^5810 = 9
unit gigit of 22^853 is equivalient to unit digit of 2^853=2

I have doubt in the last two steps,

First , Unit digit of 73^5810 = 3^ 5810 because the unit digit of 73 is 3?

and second, how can I know that when 3^5810 is expanded , its units digit will be 9?

I mean I did understand that when we see the pattern of the powers of 3 and their unit's digit, they go like this : 3, 9, 7,1,3,9,7,1

and also for 2^853 , the pattern of the powers of 2 and their units digit is 2,4,8,6,24,8,6

Quote:

Originally Posted by TheCat

Q3.Find the last digit in the expression (36472)^123! x (34767)^76!
ans: unit digit of (36472)^123! is equivalent to unit digit of 2^123!
since 123! is divisible by 4 hence unit digit of 2^123!=6

and one last doubt over here, 123! is divisible by 4 but how do u derive the units digit with that?

Thanx again

ak_shri250

2^123!: unit place is 6...
How?

2^1=2
2^2=4
2^3=8
2^4=6 {16 i.e.}
2^5=2 {32}
2^6=4 {64}
2^7=8 {128}
2^8=6 {256}

see the patern, i.e. any number with cyclicity of 4 (eg 2,3,7,

repeats after an interval of 4..

so since 123! is divisible by 4 so the whole problem reduces to 2^4 i.e.6 in its unit place....

rahul_gsp

hi
can u tell how is 123! divisble by 4 which concept is being used here

**Neo2000**

Define n!
n*(n-1)*(n-2)*.........1

So 123! = 123*122*121*....100*...............

What's the rule for a number to be divisble by 4??
Ans:The Last 2 digits have to be divisible by 4

Any number multiplied by 100 will end in 00 making it divisible by 4

Yay!!!

HillView

coz 123! = 1.2.3.4.. ...... ..120.121.122.123
So u can see that it will be divisible by 4.

indian_girl · 12:57 PM

Thanx a lot all, for replying and providing with the solutions for the problems
I am now ready with my next set of problems,

But before that , I had like to tell u all, Now that all the problems are solved by u people,
I had like to shorten this thread by uploading a .txt file containing the solutions for the problems I mentioned at the beginning of this thread ( Set 1) , such that everyone who requires the solutions for them can download the file and will not have to go through too may pages or posts, And I also request u people to just check if the solutions I have provided in that file are correct or not,( specially question no.3 of Set 1) , so that I can make a correction if any .

my next set of problem are as follows:
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Set 2:
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Q.1 Which of the following can never be in the ending of a perfect square?

a. 6
b. 00
c. 000
d. 1

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Q.2 Which of the following is not a perfect square?

a. 1,00,856
b. 3,25,137
c. 9,45,729
d All of these

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Q.3 Find the remainder when 9^100 is divided by 8?

a. 1
b. 2
c. 0
d. 4
Note: 9^100 implies 9 to the power of 100 and not 9 multiplied by 100 ...same applies to all.
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Q. 4 Find the remainder of 2^1000 when divided by 3?
a. 1
b. 2
c. 4
d. 6

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Q. 5 Find the L.C.M of (x+3) (6x^2+5x+4) and (2x^2+ 7x +3) (x+3)

a. (2x+1) (x+3) (3x+4)
b. (4x^2- 1) ( x+3)^2 ( 3x+4)
c. (4x^2- 1) ( x+3) (3x+4)
d. ( 2x-1) (x+3) (3x+4)

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Q.6 Some birds settled on the branches of a tree. First, they sat one to branch and there was one bird too many. Next they sat two to a branch and there was one branch too many. How many branches were there?

a. 3
b. 4
c. 5
d. 6

Note: I dint understand the question itself, it sounds tricky to me

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Q. 7 The square of a number greater than 1000 that is not divisible by three, leaves a remainder of

a. 1 always
b. 2 always
c. either 1 or 2
d. Cannot be said

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Q.8 If A= (-3/4)^3, B = (-2/5)^2 , C = (0.3)^2,
D =(-1.2)^2

a. A> B > C > D
b. D >A >B > C
c. D> B> C> A
d. D> C> A>B

------------------------------------------------------------------------------------Q.9 If 2 < x < 4 and 1 < y < 3 , then find the ratio of the upper limit for x+y and the lower limit of x-y

a. 6
b.7
c. 8
d. None of these
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Q.10 Find the value of x in√x+2√x+2√x+2√3x = x

a. 1
b. 3
c. 6
d. 12

Note : Here In √x+2√x+2√x+2√3x = x ,
it doesnt imply .......
(x+2 )1/2 * (x+2)1/2 * (x+2) * 3x 1/2

insted it implies .......
(x+2 (x+2 (x+2 (3x) 1/2)1/2)1/2)1/2

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Thank you