A natural number m is called a perfect square number if it can be expressed as
where n is also a natural number. For example, 4 is a perfect square number because it is a natural number and it can be expressed as
and 2 is also a natural number.
Can we write all the perfect square numbers from 1 to 100. Certainly, we can…
So, all the perfect square numbers from 1 to 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Similarly, we can keep writing perfect square numbers greater than 100.
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and so on….
Properties of Perfect Square Numbers:
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- Note that all the perfect square numbers end with 0, 1, 4, 5, 6 or 9. Can you think of any perfect square number which ends with 2, 3, 7 or 8? There is no such number. Note that perfect square numbers end with 0, 1, 4, 5, 6 or 9 but every number which ends with 0, 1, 4, 5, 6 or 9 is not a perfect square numbers. For example, 11 is not a perfect square number and it ends with 1. It cannot be expressed as
where n is a natural number. Can you write five more numbers which end with 0, 1, 4, 5, 6 or 9 and are not perfect square numbers?
- 2n non-perfect square numbers lie between two consecutive perfect square numbers. Please go through article for more details.
- Every nth perfect square number can be expressed as sum of first n odd numbers. For example, 2nd perfect square number which is 4 can be expressed as sum of first two odd numbers (1+3=4). Similarly, 5th perfect square number which is 25 can be expressed as sum of first 5 odd numbers (1+3+5+7+9=25). This point can also be re-written as sum of first n odd numbers is equal to
- Note that all the perfect square numbers end with 0, 1, 4, 5, 6 or 9. Can you think of any perfect square number which ends with 2, 3, 7 or 8? There is no such number. Note that perfect square numbers end with 0, 1, 4, 5, 6 or 9 but every number which ends with 0, 1, 4, 5, 6 or 9 is not a perfect square numbers. For example, 11 is not a perfect square number and it ends with 1. It cannot be expressed as
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