A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM. The immediate next time that they will beep together is ________

The correct option is 10.08 AM.

To find the next time both watches beep together, we need to find the least common multiple (LCM) of their individual beeping intervals. Watch X beeps every 30 seconds, and watch Y beeps every 32 seconds. The time interval after which they beep together again will be the LCM of 30 and 32 seconds.

First, let us find the prime factorization of both numbers:

$$30=2\times 3\times 5$$

$$32=2^5$$

The Least Common Multiple (LCM) is calculated by taking the highest power of each prime factor that appears in these factorizations:
$$\text{LCM}(30,32)=2^5\times 3\times 5$$
$$\text{LCM}(30,32)=32\times 3\times 5=480\text{ seconds}$$

Now, let's convert this duration from seconds into minutes:
$$\text{Time in minutes}=\frac{480}{60}=8\text{ minutes}$$

Since they beeped together at 10.00 AM, the immediate next time they will beep together is 8 minutes later. Adding 8 minutes to 10.00 AM gives 10.08 AM.