What will be the “one’s digit” in the square of the following numbers?
(i) 1234 (ii) 26387 (iii) 52698
(iv) 99880 (v) 21222 (vi) 9106
(i) ∵ Ending digit = 4 and 42 = 16
∴ (1234) will have 6 as the one’s digit.
(ii) ∵ Ending digit is 7 and 72 = 49
∴ (26387)2 will have 9 as the one’s digit.
(iii) ∵ Ending digit is 8, and 82 = 64
∴ (52692)2 will end in 4.
(iv) ∵ Ending digit is 0.
∴ (99880)2 will end in 0.
(v) ∵ 22 = 4
∴ Ending digit of (21222)2 is 4.
(vi) ∵ 62 = 36
∴ Ending digit of (9106)2 is 6.
How many natural numbers lie between 92 and 102? Between 112 and 122?
(a) Between 92 and 102
Here, n = 9 and n + 1 = 10
∴ Natural number between 92 and 102 are (2 × n) or 2 x 9, i.e. 18.
(b) Between 112 and 122
Here, n = 11 and n + 1 = 12
∴ Natural numbers between 112 and 122 are (2 × n) or 2 × 11, i.e. 22.
The square of which of the following numbers would be an odd number/an even number?
Why?
(i) 727 (ii) 158 (iii) 269 (iv) 1980
(i) 727
Since 727 is an odd number.
∴ It square is also an odd number.
(ii) 158
Since 158 is an even number.
∴ Its square is also an even number.
(iii) 269
Since 269 is an odd number.
∴ Its square is also an odd number.
(iv) 1980
Since 1980 is an even number.
∴ Its square is also an even number.
What will be the number of zeros in the square of the following numbers?
(i) 60 (ii) 400
(i) In 60, number of zero is 1
∴ Its square will have 2 zeros.
(ii) ∵ There are 2 zeros in 400.
∴ Its square will have 4 zeros.
Property 6. The difference between the squares of two consecutive natural numbers is equal to the sum of the two numbers.
Property 7. There are 2n non-perfect square numbers between the squares of the numbers n and n + 1.
How many non-square numbers lie between the following pairs of numbers:
(i) 1002 and 1012 (ii) 902 and 912
(iii) 10002 and 10012
(i) Between 1002 and 1012
Here, n=100 ∴ n x2 = 100 x 2 = 200
∴ 200 non-square numbers lie between 1002 and 1012 .
(ii) Between 902 and 912
Here, n=90 ∴ 2 x n = 2 x 90 or 180
∴ 180 non-square numbers lie between 90 and 91.
(iii) Between 10002 and 10012
Here, n = 1000 ∴ 2 x n = 2 x 1000 or 2000
∴ 2000 non-square lie between 10002 and 10012.