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 23^{2} = 529

Hence, number to be added to 525 = 23^{2} - 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 42^{2} = 1764

Hence, number to be added to 1750 = 42^{2}– 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 16^{2} = 256

Hence, number to be added to 252 = 16^{2} - 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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