Fill in the blanks:
(i) A number ending in ..... is never a perfect square.
(ii) If a number has digits ..... in the unit's place, then its square ends in 1.
(iii) Sum of first 10 odd natural numbers is .....
(iv) Number of non-square numbers between 112 and 122 is .....
(v) Number of zeros in the end of the square of 400 is .....
(vi) Square of any ..... number can be expressed as the sum of two consecutive natural numbers.
(vii) For a natural number m > 1, (2m, m2 - 1, m2 + 1) is called .....
Answer
(i) According to the property of square numbers, a number having 2, 3, 7 or 8 at its unit place is never a square number.
Hence, a number ending in 2, 3, 7 or 8 is never a perfect square.
(ii) We know that 12 = 1 and 92 = 81, so both 1 and 9 give 1 at the unit's place of the square.
Hence, if a number has digits 1 or 9 in the unit's place, then its square ends in 1.
(iii) As we know that, the sum of first n odd natural numbers = n2.
Sum of first 10 odd natural numbers = 102
= 100
Hence, sum of first 10 odd natural numbers is 100.
(iv) We know that there are 2n non-square numbers between the squares of two consecutive numbers n and (n + 1).
Here, n = 11
Non-square numbers between 112 and 122 = 2 × 11
= 22
Hence, number of non-square numbers between 112 and 122 is 22.
(v) According to the property of square numbers, if a number ends with n zeros, its square ends with 2n zeros.
The number 400 ends with 2 zeros, so its square ends with 2 × 2 = 4 zeros.
Hence, number of zeros in the end of the square of 400 is four.
(vi) We can express the square of any odd number greater than 1 as the sum of two consecutive natural numbers.
Hence, square of any odd number can be expressed as the sum of two consecutive natural numbers.
(vii) For any natural number m > 1, (2m)2 + (m2 - 1)2 = (m2 + 1)2.
Hence, for a natural number m > 1, (2m, m2 - 1, m2 + 1) is called Pythagorean triplet.
State whether the following statements are true (T) or false (F):
(i) All natural numbers are not perfect squares.
(ii) A perfect square can never be expressed as the product of pairs of equal prime factors.
(iii) A number having 2, 3, 7 or 8 at its unit place is never a square number.
(iv) A number having 0, 1, 4, 5, 6 or 9 at its unit place is always a square number.
(v) A number ending in an even number of zeros is always a perfect square.
(vi) Square of an odd number is always an odd number.
(vii) There are 2n non-square numbers between the squares of consecutive numbers n and (n + 1).
(viii) (4, 6, 8) is a Pythagorean triplet.
Answer
(i) The natural numbers 2, 3, 5, 6, 7, 8, ..... are not perfect squares.
Hence, the given statement is true.
(ii) A perfect square can always be expressed as the product of pairs of equal prime factors.
Hence, the given statement is false.
(iii) A square number always ends with 0, 1, 4, 5, 6 or 9 at its unit's place.
Hence, the given statement is true.
(iv) A square number always ends with 0, 1, 4, 5, 6 or 9 at its unit's place but the converse is not true.
For example, 30, 21, 34, 45, 76 and 89 are not square numbers.
Hence, the given statement is false.
(v) A square number can only have an even number of zeros at the end but the converse is not true.
For example, 300, 60000 and 8000000 end with an even number of zeros but these are not square numbers.
Hence, the given statement is false.
(vi) Squares of even natural numbers are even and squares of odd natural numbers are odd.
Hence, the given statement is true.
(vii) There are 2n non-square numbers between the squares of two consecutive numbers n and (n + 1).
Hence, the given statement is true.
(viii) 42 + 62
= 16 + 36
= 52
and 82 = 64
Since, 42 + 62 ≠ 82
Hence, the given statement is false.