Show that each of the following numbers is a perfect square. Also find the number whose square is the given number in each case:
(i) 1156
(ii) 2025
(iii)14641
(iv) 4761
(i) 1156
Resolving 1156 into prime factors we get,
1156 = 2 × 2 × 17 × 17
Now, grouping the factors into pairs of equal factors
We get,
1156 = (2 × 2) × (17 × 17)
As all factors are paired
Hence, 1156 is a perfect square
Again,
1156 = (2 × 17) × (2 × 17)
= 34 × 34
= (34)2
Thus, 1156 is a square of 34
(ii) 2025
Resolving 2025 into prime factors we get,
2025 = 3 × 3 × 3 × 3 × 5 × 5
Now, grouping the factors into pairs of equal factors
We get,
2025 = (3 × 3) × (3 × 3) × (5 × 5)
As all factors are paired
Hence, 2025 is a perfect square
Again,
2025 = (3 × 3 × 5) × (3 × 3 × 5)
= 45 × 45
= (45)2
Thus, 2025 is a square of 45
(iii)14641
Resolving 14641 into prime factors we get,
14641 = 11 × 11 × 11 × 11
Now, grouping the factors into pairs of equal factors
We get,
14641 = (11 × 11) × (11 × 11)
As all factors are paired
Hence, 14641 is a perfect square
Again,
14641 = (11 × 11) × (11 × 11)
= 121 × 121
= (121)2
Thus, 14641 is a square of 121
(iv) 4761
Resolving 4761 into prime factors we get,
4761 = 3 × 3 × 23 × 23
Now, grouping the factors into pairs of equal factors
We get,
4761 = (3 × 3) × (23 × 23)
As all factors are paired
Hence, 4761 is a perfect square
Again,
4761 = (3 × 23) × (3 × 23)
= 69 × 69
= (69)2
Thus, 4761 is a square of 69