1. Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057 (ii) 23453 (iii) 7928
(iv) 222222 (v) 1069 (vi) 2061
Write five numbers which you can decide by looking at their one’s digit that they are not square numbers.
(i) 1057
∵ The ending digit is 7 (which is not one of 0, 1, 4, 5, 6 or 9)
∴ 1057 cannot be a square number.
(ii) 23453
∵ The ending digit is 3 (which is not one of 0, 1, 4, 5, 6 and 9)
∴ 23453 cannot be a square number.
(iii) 7928
∵ The ending digit is 8 (which is not one of 0, 1, 4, 5, 6 and 9)
∴ 7928 cannot be a square number.
(iv) 222222
∵ The ending digit is 2 (which is not one of 0, 1, 4, 5, 6 or 9)
∴ 222222 cannot be a square number,
(v) 1069
∵ The ending digit is 9.
∴ It may or may not be a sqaure number.
Also, 30 x 30 = 900
31 x 31 = 691
32 x 32 = 1024
33 x 33 = 1089
i.e. No natural number between 1024 and 1089 is a square number.
∴ 1069 cannot be a square number.
(vi) 2061
∵ The ending digit is 1
∴ It may or may not be a square number.
∵ 45 × 45 = 2025
and 46 × 46 = 2116
i.e. No natural number between 2025 and 2116 is a square number.
∴ 2061 is not a square number.
We can write many numbers which do not end with 0, 1, 4, 5, 6 or 9. (i.e. which are not square number). Five such numbers can be:
1234, 4312, 5678, 87543, 1002007.
Which of 1232, 772 822, 1612, 1092 would end with digit 1?
The squares of those numbers end in 1 which end in either 1 or 9. The squares of 161 and 109 would end in 1.
Property 3. When a square number ends in 6, then the number whose square it is, will have 4 or 6 in its unit place.
Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s place) whether they are square numbers or not.
Any natural number ending in 0, 1, 4, 5, 6 or 9 can be or cannot be a square-number. Five such numbers are:
56790, 3671, 2454, 76555, 69209
Property 2. If a number has 1 or 9 in the unit’s place, then its square ends in 1.
For example:
(1)2 = 1, (9)2 = 81, (11)2 = 121, (19)2 = 361, (21)2 = 441.
Find the perfect square numbers between (i) 30 and 40 and (ii) 50 and 60.
Since, 1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
Thus, 36 is a perfact square number between 30 and 40.
Since, 7 x 7 = 49 and 8 x 8 = 64. it mean there is no perfect number between 49 and 64, and thus there is not perfect number between 50 and 60.
Which of the following numbers would have digit 6 at unit place.
(i) 192 (ii) 242 (iii) 262
(iv) 362 (v) 342
(i) 192: Unit’s place digit = 9
∴ 192 would not have unit’s digit as 6.
(ii) 242: Unit’s place digit = 4
∴ 242 would have unit’s digit as 6.
(iii) 262: Unit’s place digit = 6
∴ 262 would have 6 as unit’s place.
(iv) 362: Unit place digit = 6
∴ 362 would end in 6.
(v) 342: Since, the unit place digit is 4
∴ 342 would have unit place digit as 6.
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