Number System Important Questions With Solutions

Q.1  Find the minimum value of "n" such that 570x60x30x90x100x500x700x343x720x81 is perfectly divisible by " 30^n "? 

 ANS  :   

In this Question, We have to find the value of n. means highest power of  30^? in 570x60x30x90x100x500x700x343x720x81.


Step 1: We can Write 30 as 2x3x5
       30 = 2 x 3 x 5

Step 2: Now We have to search in the given number how many times 2x3x5 comes. for more ease we can also write the number as

   570 = 19 x 3 x 5 x 2
   60 = 2 x 2 x 3 x 5
   30 = 2 x 3 x 5
   90 = 2 x 3 x 3 x 5
   100 = 2 x 2 x 5 x 5
   500 = 2 x 2 x 5 x 5 x 5
   700 = 2 x 2 x 5 x 5 x 7
   343 = 7 x 7 x 7
   720 = 2 x 2 x 2 x 2 x 5 x 3 x 3
   81 = 3 x 3 x 3 x 3


Step 3: Now, we have to search for how many times 2, 3 and 5 comes

   No of 2's = 15 times
   No of 3's = 11 times
   No of 5's = 12 times


Step 4:  Now for Knowing How many times 2x3x5 comes we have to choose a minimum of the " no of 2's, no of 3's, no of 5's"

here 3 comes Least time i.e. 11

Hence the highest Power of 30 in the given Number is 11. Ans

Q.2  Find the unit digit of the product of all the prime numbers b/w 1 and (17)¹⁷ ?

 ANS : 

In this, we have to find the unit digit of the product of prime numbers between 1 to 17¹⁷


Step 1: We know Prime Number b/w 1 to 17 is

1,2,3,5,7,11,13,17


Step 2: Product of prime { according to question }

   1 x 2 x 3 x 5 x 7 x 11 x 13 x 17


Step 3: For Finding Unit digit we have to only think about the last digit (place value=ONCE)

above multiplication of Prime Number, We notice that 2 x 5 gives then unit digit 0 hence if we multiply 0 with the remaining Prime numbers then That will be also ZERO.


Hence The unit Digit is 0. Ans


Q.3  If p is the prime number, then which of the following may also be a prime number? 

(a) 2p   (b)  3p   (c)  p-3   (d)  p-2   (e)  p²

  ANS :  

We know that what Prime Number is? 
think any one prime number and put one by one to the given options.
suppose we choose 5 {prime number}

option (a) : 
2p 
= 2 x 5 
= 10  {not prime number}

option (b) :
3p
 = 3 x 5
 = 15 { again not prime}

option (c):
p-3 
 =  5- 3 
=2 {prime number}
but in this option if we put another prime number like 7 then this option failed. ( 7-3 = 4 {not prime} )

option (d):
p-2 
= 5 -2 
= 3 {prime}
if we put any prime numbers in this equation then it gona give result always Prime. 

Hence (d) is the Ans.




Q.4  "N" is the largest 3-digit number, which when divided by 3,4, and 6 leaves the remainder 1,2, and 4 respectively. what is the remainder when N is divided by 7? 


  ANS :  

According to question,

 N = largest three-digit number

⧪ if we divide N by 3 then the remainder will be 1
⧪ if we divide N by 4 then the remainder will be 2
⧪ if we divide N by 6 then the remainder will be 4

 if we divide N by 7 then remainder =? 🙆

NOW,

We know that the largest 3 digit number is 999

If we put N = 999 then,
 N/3 gives the remainder 0.    {Failed}


If we Put N = 998 then,
 N/3 gives the remainder 2.  {again Failed}


If we Put N = 994 then,
 N/3 gives the remainder 1,  {matches the condition}


 N/4 gives the remainder 2, and
 N/6 gives the remainder 4


For N = 994 all condition is satisfied, so there is no doubt N is definitely going to be 994


At last, If we Divide 994 by 7 then it will give then Remainder 0. Ans



Q.5 Find the number of zeroes in 2¹⁴⁵ x  5²³⁴ ?

  ANS :  

To find No of Zero's We have to find how many times "No of 5's & 2's {both} " comes.

Given :
 2¹⁴⁵ x  5²³⁴

It means,
     2 Comes 145 times, and
     5 comes 234 times


For every Zero, there is always 2 x 5

least ( 145, 234 ) = 145 Zero. Ans


Q.6  If  x  and y  are the two digits of the number 653XY such that this number is divisible by 40, then find 2x + y =?

  ANS :  

Given:
if N = 653XY is divisible by 40 then "2X + Y = ?"


Step 1: We can write 40 as " 8 x 5 "


Step 2: for divisible by 5,

  if 653XY is divisible by 5, then the last digit should be always 0{zero} or 5.
  So We have Find "Y = 0" or Y = 5


Step 3: For divisible by 8,

  We know, if any number is divisible by 8 then last THREE DIGIT should be divisible by 8.

here the last Three-digit is  53X should be divisible by 8 { here we have not chosen "Y" because we have already Check Y for divisibility of 5}


N = 53X

if we Put X = 6 then it is divisible by 8, it is also divisible by 8 then, here, X = 6.


Now
2X + Y = ...

For X = 6 & Y = 0,
  2x6 + 0 = 12.  Ans[1]

For X = 6 & Y = 5,

  2x6 + 5 = 17.  Ans[2]




Q.7 What is the negative remainder if 121 is divided by 5?

  ANS :  

Q.8  If  17²⁰⁰ is divided by 18 remainders are?

  ANS :  

Q.9  (49)₅ + (23)₉  = ( .... )₈

  ANS :  

Q.10  Find the digit in a hundred positions of  5² x  2¹⁰¹?

  ANS :  

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