Absolutely! Let's address each part of the question step by step:
Part (c): Determining when the four bells will next ring together.
1. Identify the intervals: The four bells ring at intervals of 8, 12, 16, and 24 minutes.
2. Find the least common multiple (LCM):
- The LCM of the intervals 8, 12, 16, and 24 is 48 minutes.
3. Calculate the start time:
- The bells initially rang together at 8:30 AM. To work with minutes, convert this time to minutes since midnight.
- 8:30 AM is [tex]\(8 \times 60 + 30 = 510\)[/tex] minutes past midnight.
4. Determine the next ring time:
- The next time the bells will ring together is 48 minutes after the initial ring.
- Adding the LCM to the initial time: [tex]\(510 + 48 = 558\)[/tex] minutes.
5. Convert back to hours and minutes:
- To convert 558 minutes back to hours and minutes: [tex]\(558 \div 60 = 9 \text{ hours and } 18 \text{ minutes}\)[/tex].
- Therefore, the bells will all ring together again at 9:18 AM.
Part (d): Finding the greatest 4-digit number exactly divisible by 15, 25, 45, and 75.
1. Identify the divisors: The numbers are 15, 25, 45, and 75.
2. Find the least common multiple (LCM):
- The LCM of 15, 25, 45, and 75 is 225.
3. Determine the greatest 4-digit number divisible by 225:
- The largest 4-digit number is 9999. We need to find the largest number less than or equal to 9999 that is divisible by 225.
- Dividing 9999 by 225 and taking the integer part gives us the quotient: [tex]\( \left\lfloor \frac{9999}{225} \right\rfloor = 44 \)[/tex].
- Multiplying the quotient by the LCM: [tex]\( 44 \times 225 = 9900 \)[/tex].
Thus, the greatest 4-digit number exactly divisible by 15, 25, 45, and 75 is 9900.
