Finding the Sum of Largest and Smallest 4-Digit Numbers Using Single Digit Primes

This question asks us to find the sum of two specific 4-digit numbers. These numbers must be formed using only single-digit prime numbers, and importantly, each digit must be used only once (no repetition) within each number.

Understanding Prime Numbers

First, let's recall what prime numbers are. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. The single-digit numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

• Out of these, the prime numbers are: 2, 3, 5, and 7.
• The number 1 is not considered prime.
• Numbers like 4, 6, 8, and 9 are composite (they have more than two divisors). 0 is neither prime nor composite.

So, the set of single-digit prime numbers we can use is {2, 3, 5, 7}. We need to form 4-digit numbers using each of these digits exactly once.

Constructing the Largest 4-Digit Number

To create the largest possible 4-digit number using the digits {2, 3, 5, 7} without repetition, we should arrange the digits in descending order from left to right (from the thousands place to the units place).

1. The largest digit is 7. Place it in the thousands place.
2. The next largest digit is 5. Place it in the hundreds place.
3. The next largest digit is 3. Place it in the tens place.
4. The smallest digit is 2. Place it in the units place.

Therefore, the largest 4-digit number formed is 7532.

Constructing the Smallest 4-Digit Number

To create the smallest possible 4-digit number using the same digits {2, 3, 5, 7} without repetition, we should arrange the digits in ascending order from left to right.

1. The smallest digit is 2. Place it in the thousands place.
2. The next smallest digit is 3. Place it in the hundreds place.
3. The next smallest digit is 5. Place it in the tens place.
4. The largest digit is 7. Place it in the units place.

Therefore, the smallest 4-digit number formed is 2357.

Calculating the Sum

The final step is to find the sum of the largest number (7532) and the smallest number (2357).

We can perform the addition:

$ \begin{array}{@{}c@{\,}c@{}c@{}c@{}c} & 7 & 5 & 3 & 2 \\ + & 2 & 3 & 5 & 7 \\ \hline & 9 & 8 & 8 & 9 \\ \end{array} $

Alternatively, using standard notation:

Sum = Largest Number + Smallest Number

Sum = \(7532 + 2357\)

Sum = \(9889\)

Conclusion

The sum of the largest and smallest 4-digit numbers made by using single-digit prime numbers (2, 3, 5, 7) without repetition is 9889.