Question:Without actual division show that `52563744` is divisible by `24`. 

`24=3 xx 8` are co-prime.
The sum of the digits is given number is `36`, which is divisible by `3`.
So, the given number is divisible by `3`.
The number formed by the last 3-digits of the given number is `744`, which is divisible by `8`.
So, the given number is divisible by `8`.
Thus, the given number is divisible by both `3` and `8`, where `3` and `8` are co-prime.
So, it is divisible by `3 xx 8` , i.e. `24`.

Question:Multiply `5793405` by `99999` by short cut method. 

`5793405 xx 99999` 
`= 5793405 xx (100000 - 1)`                        
`= (579340500000 - 5793405)`                        
`= 579334706595.`

Question:Evaluate: 
i) `896 xx 137 + 986 xx 863 ` 
ii) `983 xx 207 - 983 xx 107` 

i) `986 xx 137 + 986 xx 863` 
`= 986 ( 137 + 863 )`
`= 986 xx 1000`
`= 980000.`

ii) `983 xx 207 - 983 xx 107`
` = 983 xx ( 207 - 107)`
`= 983 xx 100 = 98300.`

Question:Evaluate following expressions:
 i) `( 527 xx 527 xx 527 + 183 xx 183 xx 183)/(527 xx 527 - 527 xx 183 + 183 xx 183)`
 
 ii) `(458 xx 458 xx 458 - 239 xx 239 xx 239)/(458 xx 458 xx + 458 xx 239 + 239 xx 239)`

 iii) `((614 + 168)^2 - (614 - 168)^2)/(614 xx 168)`

 iv) `((832 + 278)^2 + (832 - 278)^2)/(832 xx 832 + 278 xx 278)` 

i) Given Expression 

 `= ((527)^3 + (183)^3)/((527)^2 - 527 xx 183 + (183)^2)`

 `= (a^3 + b^3)/(a^2 - ab + b^2)`

 `= (a + b) = (527 + 183) + 710)`


 ii) Given Expression 

 `= ((458)^3-(239)^3)/((458)^2 + 458 xx 239 + (239)^2)`

 `= (a^3-b^3)/(a^2 + ab + b^2)`

 `= (a - b) = (457 - 239) = 219)`


 iii) Given Expression

 `= ((a + b)^2 - (a - b)^2)/(ab)` 

 `= (4ab)/4`

 `= 4`


 iv) Given Expression

 `= ((a + b)^2 + (a - b)^2)/(a^2 + b^2)`

 `= (2(a^2 + b^2))/(a^2 + b^2)`

 `= 2`

Question:On dividing `15968` by a certain number, the quotient is `89` and the remainder is `37`. Find the divisor. 

Divisor = `(text{Dividend - Remainder})/text{Quotient} = (15968 - 37)/89 = 179`

Question:Find the unit's digit in the product `(256 xx 27 xx 159 xx 182)`. 

Product of units the given numbers = `(6 xx 7 xx 9 xx 2) = 756`
  `:.` unit digit in the given product is `6`.

Question:2.Find the unit,s digit in the product` (3^67 xx 6^39 xx 7^53)` 

sol.Clearly ,unit digit in `3^4` is 1.
      
     :. unit digit in `3^64` is 1.
      
     :. unit digit in `3^67` is 7. [unit digit in `1 xx 3 xx 3 xx 3` is 7]
       
      clearly unit digit in every power of 6 is 6.
       
      :.unit digit in `6^39` is 6.
       
       clearly unit digit in `7^4` is 1.
     
       :. unit digit in `7^72`is 1.
       
       :. unit digit in `7^53`is 7. [unit digit in `1 xx 7` is 7]
        
       :.unit digit in given product =unit digit in`(7 xx 6 xx 7) = 4`

Question:4.which of the following are prime numbers.?
 (i)191 (ii) 241 (iii)337 (iv) 391 

Sol.(i) ,14>√191.
prime numbers less than 14 are 2.3,5,7,11,13.
191 is not divisible by any of them .
So 191 is a prime number.

(ii) Clearly,16>√241.
 prime numbers less than 16 are 2,3,5,7,11,13.
  241 is not divisible by any of them. 
:.241 is a prime number.
 (iii) Clearly,19>√337.
 prime numbers less than 19 are 2,3,5,7,11,13,17.
337 is not disivible by any of them.
:. 337 is a prime number.
(iv)Clearly,20>√391.
 prime number less than 20 are 2,3,5,7,11,13,17,19.
We find than 391 is divisible by 17.
:. 391 is not prime.

Question:5.Which of the following numbers is not disivible by 2.
 3567516,9130525,7870832,1371594 

Sol.Clearly,the unit,s digit of 913o525 is 5,which is not divisible by 2.
So,9130525 is not disivible by 2.
Each of the numbers has a unit digit disivible by 2.
So, each one other than 9130525 is disivible by 2.

Question:6.Which of the following numbers is divisible by 3.

(i)541326   (ii) 5967013 

Sol.(i) Sum of digit in 541326 is 21,which is divisible by 3.
So,541326 is divisible by 3.
  (ii) Sum of digits in 5967013 is 31,which is not divisible by 3.
  Hence,5967013 is not divisible  by 3.