MBA Nexus - CAT & OMETS 2024, [12-10-2024 08:04]
Q5 A "prime time" is defined as a digital display which indicates time only when both hours and minutes are a prime number, on a twelve hour clock. Which of the following is the number of minutes in a day on which the "prime time" has shown the time?

MBA Nexus - CAT & OMETS 2024, [12-10-2024 08:05]
[ Poll : Ans ]
- 85
- 144
- 170
- 25

To solve this problem, we need to consider the concept of "prime time" as defined in the question: both the hours and the minutes should be prime numbers on a twelve-hour clock.

### Step-by-Step Breakdown:

1. **Prime Numbers under 12**:
   Since the clock operates on a 12-hour cycle, we need the prime numbers between 1 and 12:
   - Primes less than 12: 2, 3, 5, 7, 11

   These are the possible prime numbers for the **hour** values.

2. **Prime Numbers under 60**:
   Next, we look at the minutes on the clock, which can range from 0 to 59. We need all prime numbers in this range:
   - Primes less than 60: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

   These are the possible **minute** values.

3. **Counting "Prime Time" Instances in a 12-hour Cycle**:
   For each prime hour, we can pair it with any of the prime minute values. Therefore, the total number of valid "prime time" instances in a 12-hour cycle is:

   \[
   \text{Number of prime hours} \times \text{Number of prime minutes} = 5 \times 16 = 80
   \]

4. **Multiplying for a Full Day**:
   Since a full day has two 12-hour cycles (AM and PM), we need to multiply by 2:

   \[
   \text{Total "prime time" instances in a day} = 80 \times 2 = 160
   \]

Thus, the correct number of minutes in a day where the time is "prime time" is 160 minutes.

However, **none of the provided options match exactly**, so it could be a typo, but based on the calculation, 160 is the closest logical answer.

Let me know if you'd like further details or clarifications.

### 5 Related Questions:
1. How can prime numbers be efficiently determined in a larger range?
2. What are the properties of prime numbers and how are they applied in time-related problems?
3. Can the concept of "prime time" be extended to a 24-hour clock, and how would that affect the answer?
4. How does modular arithmetic help in understanding clock problems like this?
5. How could this problem be modified if it used a 24-hour format instead?

**Tip:** Remember that prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

A "prime time" is defined as a digital display which indicates time only when both hours and minutes are a prime number, on a twelve hour clock. Which of the following is the number of minutes in a day on which the "prime time" has shown the time?

Explore how to find the total number of minutes in a day where both the hour and minute on a 12-hour clock are prime numbers. This problem involves concepts such as prime numbers and clock arithmetic, providing a detailed step-by-step solution for Grades 9-10.

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