General Properties of Squares


 
 
Concept Explanation
 

General Properties of Squares

The square numbers are the values which are produced when we multiply a number by itself. Some of the properties are:

  • Square of 1 is equal to 1
  • Square of positive numbers are positive in nature
  • Square of negative numbers is also positive in nature. For example, (-3)2 = 9
  • Square of zero is zero
  • Square of root of a number is equal to the value under the root. For example, (√3)2 = 3
  • The unit place of square of any even number will have an even number only.
  • If a number has 1 or 9 in the unit’s place, then its square ends in 1.
  • If a number has 4 or 6 in the unit’s place, then its square ends in 6.
  • Property 1:  The square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.

    In other words, a perfect square leaves remainder 0 or 1 on division by 3.

    Verification: The above property can be easily verified from the following computations:

    Square number Remainder when divided by 3
    2^{2}=4=3times1+1 1
    3^{2}=9=3times3+0 0
    4^{2}=16=3times5+1 1
    5^{2}=25=3times8+1 1
    6^{2}=36=3times12+0 0
    7^{2}=49=3times16+1 1
    8^{2}=64=3times21+1 1

    Application: If a number when divided by 3 leaves remainder 2, then it is not a perfect square.

    Property 2:  The square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.

    In other words, a perfect square leaves remainder 0 or 1 on division by 4.

    Property 3:  The square of a natural number n is equal to the sum of first n odd natural numbers.

    We have, 

                1^{2}=1= Sum of first 1 odd natural number,

                2^{2}=1+3= Sum of first 2 odd natural numbers,

                3^{2}=1+3=5= Sum of first 3 odd natural numbers.

                4^{2}=1+3=5+7= Sum of first 4 odd natural numbers.

    Property 4 The sum of first n odd natural numbers is n^{2} i.e.

                        1+3+5+7+....+(2n-1)=n^{2}

    Sample Questions
    (More Questions for each concept available in Login)
    Question : 1

    144 can be express as the sum of first ___________________ consecutive odd numbers.

    Right Option : C
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    Explanation
    Question : 2

    To find the exact square of 11 without using multiplication which property can be used ?

    Right Option : D
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    Explanation
    Question : 3

    Find the square of  sqrt{18}.

    Right Option : A
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    Explanation
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