Problem Statement
Find the greatest number that will divide 43, 91, and 183 leaving remainders 7, 9, and 15 respectively.
Explanation
The required number is the HCF of the differences between each number and its corresponding remainder. This is because the number divides these differences exactly.
Step 1: Calculate differences. 43 minus 7 equals 36. 91 minus 9 equals 82. 183 minus 15 equals 168. Step 2: Find HCF of 36, 82, and 168. Prime factorization: 36 equals 2 squared times 3 squared. 82 equals 2 times 41. 168 equals 2 cubed times 3 times 7. HCF equals 2 which is common to all. Wait, let me recalculate. HCF of 36 and 82: Using Euclid: 82 equals 36 times 2 plus 10. 36 equals 10 times 3 plus 6. 10 equals 6 times 1 plus 4. 6 equals 4 times 1 plus 2. 4 equals 2 times 2 plus 0. HCF of 36 and 82 equals 2. But with 168, HCF should be recalculated properly.
Code Solution
SolutionRead Only
// Concept: If a number leaves remainders r1, r2, r3 // when dividing n1, n2, n3, then: // Required number = HCF(n1-r1, n2-r2, n3-r3) // Given: // 43 leaves remainder 7 // 91 leaves remainder 9 // 183 leaves remainder 15 // Differences: // 43 - 7 = 36 // 91 - 9 = 82 // 183 - 15 = 168 // Find HCF(36, 82, 168): // 36 = 2² × 3² // 82 = 2 × 41 // 168 = 2³ × 3 × 7 // HCF = 2 (common factor) // But checking manually: // 36, 82, 168 // HCF(36,82) = 2 // HCF(2,168) = 2
Practice Sets
This question appears in the following practice sets:
