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Find the smallest number by which the given number must be multiplied so that the product is a perfect square:(i) 23805(ii) 12150(iii) 7688 |
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Answer» (i) 23805 The prime factors for 23805 23805 = 3×3×23×23×5 ( grouping the prime factors in equal pairs we get,) = (3×3) × (23×23) × 5 Here 5 is left out. So, multiply by 5 we get, = 23805 × 5 = (3×3) × (23×23) × (5×5) = (3×5×23) × (3×5×23) = 345 × 345 = (345)2 (ii) 12150 The prime factors for 12150 12150 = 2×2×2×2×3×3×5×5×2 (grouping the prime factors in equal pairs we get,) = (2×2) × (2×2) × (3×3) × (5×5) × 2 Here, prime factor 2 is left out. So, multiply by 2 we get, = 12150 × 2 = (2×2) × (2×2) × (3×3) × (5×5) × (2×2) = (2×2×3×5×2) × (2×2×3×5×2) = 120 × 120 = (120)2 (iii) 7688 The prime factors for 7688 7688 = 2×2×31×31×2 (grouping the prime factors in equal pairs we get,) = (2×2) × (31×31) × 2 Here, prime factor 2 is left out. So, multiply by 2 we get, = 7688 × 2 = (2×2) × (31×31)× (2×2) = (2×31×2) × (2×31×2) = 124 × 124 = (124)2 |
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