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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 |
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Answer» (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. |
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