Problem Statement
Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they toll together again?
Explanation
When events occur at regular intervals, they occur together at intervals equal to the LCM of their individual periods. We need to find LCM of 9, 12, and 15.
Prime factorization: 9 equals 3 squared. 12 equals 2 squared times 3. 15 equals 3 times 5. LCM equals product of highest powers of all prime factors equals 2 squared times 3 squared times 5 equals 4 times 9 times 5 equals 180 minutes. This means after 180 minutes, all three bells will toll together for the first time after the initial tolling.
Code Solution
SolutionRead Only
// Intervals: 9, 12, 15 minutes // Prime Factorization: // 9 = 3² // 12 = 2² × 3 // 15 = 3 × 5 // LCM = Highest power of all primes // LCM = 2² × 3² × 5 // LCM = 4 × 9 × 5 // LCM = 180 minutes // Verification: // Bell 1: 9, 18, 27, ..., 180 // Bell 2: 12, 24, 36, ..., 180 // Bell 3: 15, 30, 45, ..., 180 // All meet at 180 minutes
Practice Sets
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