4. Three bells ring at intervals of 16 minutes, 24 minut 30 minutes. After how much time will they ri together again, if they start ringing simultaneously

To find out when the three bells will ring together again, we need to determine the least common multiple (LCM) of their ringing intervals: 16 minutes, 24 minutes, and 30 minutes. 1. **Find the prime factorization of each number:** - \( 16 = 2^4 \) - \( 24 = 2^3 \times 3^1 \) - \( 30 = 2^1 \times 3^1 \times 5^1 \) 2. **Identify the highest power of each prime factor:** - For \( 2 \): the highest power is \( 2^4 \) (from 16) - For \( 3 \): the highest power is \( 3^1 \) (from both 24 and 30) - For \( 5 \): the highest power is \( 5^1 \) (from 30) 3. **Calculate the LCM:** \[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \] \[ = 16 \times 3 \times 5 \] \[ = 48 \times 5 \] \[ = 240 \] Thus, the least common multiple of 16, 24, and 30 is 240 minutes. Therefore, the three bells will ring together again after **240 minutes**, which is equivalent to **4 hours**.