The digits 1,2,3,4,5,6,7,8,9 are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes
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To find the sum of the four two-digit prime numbers formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, we need to consider all the possible combinations.
Let's start by finding all the two-digit prime numbers that can be formed using the given digits:
1. The first digit of a prime number cannot be 1 because it would result in a single-digit number (e.g., 12, 13, 14). So, we start with the first digit being 2.
- A prime number starting with 2: 23, 29
- Remaining digits: 1, 4, 5, 6, 7, 8, 9
2. Next, we consider prime numbers starting with 3:
- A prime number starting with 3: 37
- Remaining digits: 1, 4, 5, 6, 7, 8, 9
3. Now, we move on to prime numbers starting with 5:
- A prime number starting with 5: 53
- Remaining digits: 1, 4, 6, 7, 8, 9
4. Lastly, we look at prime numbers starting with 7:
- A prime number starting with 7: 79
- Remaining digits: 1, 4, 6, 8, 9
Now, we have the four two-digit prime numbers: 23, 29, 37, 53, 79.
To find their sum:
23 + 29 + 37 + 53 + 79 = 221
Therefore, the sum of the four two-digit prime numbers formed is 221.
To find the sum of the four prime numbers, we first need to determine what those four primes are.
Let's start with the first two-digit prime. The prime numbers between 10 and 99 are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Since we are given that each digit can only be used once, we can rule out any prime numbers that have repeated digits.
Let's start with the first digit in the first two-digit prime number. It can only be 1, 2, 3, 4, 5, 6, 7, 8, or 9. The second digit can only be one of the remaining digits from the set.
We can try each possibility until we find a two-digit prime number. Let's begin:
- If the first digit is 1:
- The second digit can be 2, 3, 4, 5, 6, 7, 8, or 9.
- However, combining 12 gives us a non-prime number.
- Let's try the other possibilities.
- Combining 13 gives us the prime number 13.
- Combining 14 gives us a non-prime number.
- Combining 15 gives us a non-prime number.
- Combining 16 gives us the prime number 61.
- Combining 17 gives us a non-prime number.
- Combining 18 gives us a non-prime number.
- Combining 19 gives us the prime number 19.
So, the first two-digit prime number can be 13, 61, or 19.
Next, we need to consider the remaining digits, as they will determine the second two-digit prime number.
- If the first two-digit prime number is 13:
- The remaining available digits are 2, 4, 5, 6, 7, 8, and 9.
- We need to find a two-digit prime number using these digits.
- Trying all possible combinations, we find that 23 is a prime number.
- So, the second two-digit prime number, given that the first is 13, is 23.
Next, we need to consider the remaining digits to determine the third two-digit prime number.
- If the first two-digit prime number is 61:
- The remaining available digits are 2, 3, 4, 5, 7, 8, and 9.
- Trying all possible combinations, we find that 67 is a prime number.
- So, the second two-digit prime number, given that the first is 61, is 67.
Finally, there is only one possibility left for the first two-digit prime number (19). Therefore, the last two-digit prime number is determined by the remaining digits.
- If the first two-digit prime number is 19:
- The remaining available digits are 2, 3, 4, 5, 6, 7, and 8.
- Trying all possible combinations, we find that 89 is a prime number.
- So, the second two-digit prime number, given that the first is 19, is 89.
Now, we have our four two-digit prime numbers: 13, 23, 67, and 89.
To find the sum of these primes, we add them together:
13 + 23 + 67 + 89 = 192.
Therefore, the sum of these four prime numbers is 192.