Question 1:Which of the following numbers are perfect squares?
(i) 484
(ii) 625
(iii) 576
(iv) 941
(v) 961
(vi) 2500
Answer 4:
For each question, factorise the number into its prime factors.
(i) 14283 = 3 x 3 x 3 x 23 x 23
Grouping the factors into pairs:
14283 = (3 x 3) x (23 x 23) x 3
Here, the factor 3 does not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 14283 must be divided for it to be a perfect square is 3.
(ii) 1800= 2 x 2 x 2 x 3 x 3 x 5 x 5
Grouping the factors into pairs:
1800 = (2 x 2) x (3 x 3) x (5 x 5) x 2
Here, the factor 2 does not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 1800 must be divided for it to be a perfect square is 2.
(iii) 2904 = 2 x 2 x 2 x 3 x 11 x 11
Grouping the factors into pairs:
2904 = (2 x 2) x (11 x 11) x 2 x 3
Here, the factors 2 and 3 do not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 2904 must be divided for it to be a perfect square is 2 x 3, i.e. 6.
Question 5:Which of the following numbers are perfect squares?
11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121
Answer 5:
11: The perfect squares closest to 11 are 9 (9 = 3^{2}) and 16 (16 = 4^{2}). Since 3 and 4 are consecutive numbers, there are no perfect squares between 9 and 16, which means that 11 is not a perfect square.
12: The perfect squares closest to 12 are 9 (9 =3^{2}) and 16 (16 = 4^{2}). Since 3 and 4 are consecutive numbers, there are no perfect squares between 9 and 16, which means that 12 is not a perfect square.
16 = 4^{2}
32: The perfect squares closest to 32 are 25 (25 = 5^{2}) and 36 (36 = 6^{2}). Since 5 and 6 are consecutive numbers, there are no perfect squares between 25 and 36, which means that 32 is not a perfect square.
36 = 6^{2}
50: The perfect squares closest to 50 are 49 (49 = 7^{2}) and 64 (64 = 8^{2}). Since 7 and 8 are consecutive numbers, there are no perfect squares between 49 and 64, which means that 50 is not a perfect square.
64 = 82
79: The perfect squares closest to 79 are 64 (64 = 8^{2}) and 81 (81 = 9^{2}). Since 8 and 9 are consecutive numbers, there are no perfect squares between 64 and 81, which means that 79 is not a perfect square.
81 = 9^{2}
111: The perfect squares closest to 111 are 100 (100 = 10^{2}) and 121 (121 = 11^{2}). Since 10 and 11 are consecutive numbers, there are no perfect squares between 100 and 121, which means that 111 is not a perfect square.
121 = 112.
Question 6:Using prime factorization method, find which of the following numbers are perfect squares?
189, 225, 2048, 343, 441, 2916, 11025, 3549
Answer 6:
(i) 189 = 3 x 3 x 3 x 7
Grouping them into pairs of equal factors:
189 = (3 x 3) x 3 x 7
The factors 3 and 7 cannot be paired. Hence, 189 is not a perfect square.
(ii) 225 = 3 x 3 x 5 x 5
Grouping them into pairs of equal factors:
225 = (3 x 3) x (5 x 5)
There are no left out of pairs. Hence, 225 is a perfect square.
(iii) 2048 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Grouping them into pairs of equal factors:
2048 = (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2) x 2
The last factor, 2 cannot be paired. Hence, 2048 is not a perfect square.
(iv) 343 = 7 x 7 x 7
Grouping them into pairs of equal factors:
343 = (7 x 7) x 7
The last factor, 7 cannot be paired. Hence, 343 is not a perfect square.
(v) 441 = 3 x 3 x 7 x 7
Grouping them into pairs of equal factors:
441 = (3 x 3) x (7 x 7)
There are no left out of pairs. Hence, 441 is a perfect square.
(vi) 2916 = 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3
Grouping them into pairs of equal factors:
2916 = (2 x 2) x (3 x 3) x (3 x 3) x (3 x 3)
There are no left out of pairs. Hence, 2916 is a perfect square.
(vii) 11025 = 3 x 3 x 5 x 5 x 7 x 7
Grouping them into pairs of equal factors:
11025 = (3 x 3) x (5 x 5) x (7 x 7)
There are no left out of pairs. Hence, 11025 is a perfect square.
(viii) 3549 = 3 x 7 x 13 x 13
Grouping them into pairs of equal factors:
3549 = (13 x 13) x 3 x 7
The last factors, 3 and 7 cannot be paired. Hence, 3549 is not a perfect square.
Hence, the perfect squares are 225, 441, 2916 and 11025.
Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 