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
In each problem, factorise the number into its prime factors.
(i) 1156 = 2 x 2 x 17 x 17
Grouping the factors into pairs of equal factors, we obtain:
1156 = (2 x 2) x (17 x 17)
No factors are left over. Hence, 1156 is a perfect square. Moreover, by grouping 1156 into equal factors:
1156 = (2 x 17) x (2 x 17)
= (2 x 17)2
Hence, 1156 is the square of 34, which is equal to 2 x 17.
(ii) 2025 = 3 x 3 x 3 x 3 x 5 x 5
Grouping the factors into pairs of equal factors, we obtain:
2025 = (3 x 3) x (3 x 3) x (5 x 5)
No factors are left over. Hence, 2025 is a perfect square. Moreover, by grouping 2025 into equal factors:
2025 = (3 x 3 x 5) x (3 x 3 x 5)
= (3 x 3 x 5)2
Hence, 2025 is the square of 45, which is equal to 3 x 3 x 5.
(iii) 14641 = 11 x 11 x 11 x 11
Grouping the factors into pairs of equal factors, we obtain:
14641 = (11 x 11) x (11 x 11)
No factors are left over. Hence, 14641 is a perfect square. The above expression is already grouped into equal factors:
14641 = (11 x 11) x (11 x 11)
= (11 x 11)2
Hence, 14641 is the square of 121, which is equal to 11 x 11.
(iv) 4761 = 3 x 3 x 23 x 23
Grouping the factors into pairs of equal factors, we obtain:
4761 = (3 x 3) x (23 x 23)
No factors are left over. Hence, 4761 is a perfect square. The above expression is already grouped into equal factors:
4761 = (3 x 23) x (3 x 23)
= (3 x 23)2
Hence, 4761 is the square of 69, which is equal to 3 x 23.