Find the least number which must be added to each of the following numbers so asto get a perfect square. Also find the square root of the perfect square so obtained

(i) 525 (ii) 1750


(iii) 252 (iv) 1825


(v) 6412


(i) The square root of 525 can be calculated by long division method as:



22



2



5̅ 2̅5̅


-4



42



125


84




41



The remainder is 41


It represents that the square of 22 is less than 525


Next number is 23 and 232 = 529


Hence, number to be added to 525 = 232 - 525


= 529 - 525


= 4


The required perfect square is 529 and = 23


(ii) The square root of 1750 can be calculated by long division method as follows:



41



4



1̅7̅ 5̅0̅


-16



81



150


81




69



The remainder is 69


It represents that the square of 41 is less than 1750


The next number is 42 and 422 = 1764


Hence, number to be added to 1750 = 422– 1750


= 1764 - 1750


= 14


The required perfect square is 1764 and = 42


(iii) The square root of 252 can be calculated by long division method as follows:



15



1



2̅ 5̅2̅


-1



25



152


125




27



The remainder is 27. It represents that the square of 15 is less than 252


The next number is 16 and 162 = 256


Hence, number to be added to 252 = 162 - 252


= 256 - 252


= 4


The required perfect square is 256 and = 16


(iv) The square root of 1825 can be calculated by long division method as follows:



42



4



1̅8̅ 2̅5̅


-16



82



225


164




61



Here, 82 * 2 = 164 less than 225


Required difference = 249 – 225


= 24


Hence, number to be added to 1825 = 1825 + 24


= 1849


The required perfect square is 256 and = 43


(v) The square root of 6412 can be calculated by long division method as follows:




8


8



6412


64



16



12



Required difference = 161 – 12


= 149


Hence, number to be added to 6412 = 6412 + 149


= 6561


The required perfect square is 6561 and = 81


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