Real Numbers

⭐ Important Questions

1. Use Euclid's division algorithm to find HCF of 4052 and 12576.
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   12576 = 4052 x 3 + 420\n4052 = 420 x 9 + 272\n420 = 272 x 1 + 148\n272 = 148 x 1 + 124\n148 = 124 x 1 + 24\n124 = 24 x 5 + 4\n24 = 4 x 6 + 0\nHCF = 4.
2. Prove that √5 is irrational.
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   Assume √5 = ab in lowest terms. Then 5 = a²b² => a² = 5b². So 5 divides a² => 5 divides a. Let a = 5c. Then 25c² = 5b² => 5c² = b² => 5 divides b. Contradiction since a,b are co-prime. Hence √5 is irrational.
3. Find LCM and HCF of 510 and 92, verifying HCF x LCM = product.
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   510 = 2 x 3 x 5 x 17\n92 = 2² x 23\nHCF = 2\nLCM = 2² x 3 x 5 x 17 x 23 = 23460\nProduct = 510 x 92 = 46920\nHCF x LCM = 2 x 23460 = 46920. Verified.