General Mathematics

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

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Question:Multiply `5793405` by `99999` by short cut method. 
Answer`5793405 xx 99999` 
`= 5793405 xx (100000 - 1)`                        
`= (579340500000 - 5793405)`                        
`= 579334706595.`

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

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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)` 
Answeri) 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`

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Question:On dividing `15968` by a certain number, the quotient is `89` and the remainder is `37`. Find the divisor. 
AnswerDivisor = `(text{Dividend - Remainder})/text{Quotient} = (15968 - 37)/89 = 179`

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

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

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

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