Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again for the first time?

To solve this problem, we need to find the least common multiple (LCM) of 50 seconds and 48 seconds. The LCM of two numbers is the smallest number that is a multiple of both numbers.

Step 1: Prime factorize the two numbers.

• Prime factors of 50 = 2 * 5 * 5
• Prime factors of 48 = 2 * 2 * 2 * 2 * 3

Step 2: Find the LCM.

• LCM = product of the highest powers of all the factors that appear in either number.
• LCM = 2 * 2 * 2 * 2 * 3 * 5 * 5 = 2400 seconds

So, the two alarm clocks will beep together again after 2400 seconds.

Step 3: Convert the seconds back into hours and minutes.

• There are 60 seconds in a minute, so 2400 seconds = 2400/60 = 40 minutes.

Therefore, if the two alarm clocks first beep together at 12 noon, they will beep together again at 12:40 PM.

To solve this problem, we need to find the least common multiple (LCM) of 50 seconds and 48 seconds. The LCM of two numbers is the smallest number that is a multiple of both numbers.

Step 1: Prime factorization of both numbers

• 50 = 2 * 5 * 5
• 48 = 2 * 2 * 2 * 2 * 3

Step 2: Find the LCM The LCM is found by multiplying the highest power of all the factors that appear in either number. So, LCM = 2^4 * 3 * 5^2 = 16 * 3 * 25 = 1200 seconds

Step 3: Convert the LCM from seconds to minutes 1200 seconds = 20 minutes

So, the two alarm clocks will beep together again 20 minutes after 12 noon, which is at 12:20 PM.