Consider a digital clock. The first time after midnight when the three digits are all prime numbers is 2:22 am. What is the last time before 10:00 am, when the three digits on the clock are all prime numbers?
Details and assumptions
If you determine that the time is 4:23 am, type in 423 as your numerical answer.
757
To find the last time before 10:00 am when the three digits on the clock are all prime numbers, we can start by listing out all the prime numbers between 1 and 9.
The prime numbers between 1 and 9 are:
2, 3, 5, and 7.
Since the hours on a digital clock can range from 1 to 12, we can see that the last possible hour with all prime digits is 7.
Now, let's consider the possible minutes for each hour. The minutes on a digital clock can range from 00 to 59.
For the first digit of the minutes, it can be any digit from 0 to 5.
For the second digit of the minutes, it can be any digit from 0 to 9.
Combining all the possibilities, the last time before 10:00 am when the three digits on the clock are all prime numbers is 7:59 am.
Therefore, the numerical answer is 759.
To solve this problem, we need to find the last time before 10:00 am when the three digits on the clock are all prime numbers. Let's break down the problem into smaller steps:
Step 1: Determine the prime numbers from 0 to 9.
- Prime numbers are numbers greater than 1 that are divisible only by 1 and themselves.
- The prime numbers from 0 to 9 are 2, 3, 5, and 7.
Step 2: Analyze the format of the time on a digital clock.
- A digital clock displays time in the format of HH:MM, where HH represents the hours and MM represents the minutes.
Step 3: Determine the possible values for each digit.
- For the hours (HH), there are two digits to be considered.
- For the minutes (MM), there are three digits to be considered.
- The possible values for each digit are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Step 4: Construct a table of possible time combinations.
- Start with the first digit of the hours (HH) and iterate through all possible values, from 0 to 2.
- For each value of the first digit, iterate through all possible values for the second digit, from 0 to 9.
- Repeat this process for the first digit of the minutes (MM), iterated through from 0 to 5.
- For the second and third digits of the minutes, iterate through all possible values from 0 to 9.
Step 5: Identify the last time before 10:00 am when all three digits are prime.
- Check each time combination from the table and determine if all three digits are prime.
- If a combination meets the criteria, continue checking for subsequent combinations until we reach 10:00 am.
- Once we reach 10:00 am, identify the last qualifying combination before this time.
Step 6: Convert the time to its numerical representation.
- Take the hours (HH) and multiply by 100 to convert them into the first two digits of a three-digit number.
- Add the minutes (MM) to the resulting number to obtain the three-digit numerical representation of the time.
Following these steps, we can find the last time before 10:00 am when the three digits on the clock are all prime numbers.