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Question

Difficulty: MediumClassification of Numbers

Identify all two-digit prime numbers where both the tens digit and the units digit are strictly prime numbers. What is the sum of the largest and the smallest numbers that meet this criterion?

Answer: 96

Answer

96
The correct answer requires finding the intersection of two distinct sets: two-digit numbers formed entirely by prime digits, and two-digit numbers that are mathematically prime. This resulting set is {23, 37, 53, 73}. Adding the minimum value (23) and maximum value (73) yields 96.

Step-by-Step Solution

1
Identify valid single digits.
The single-digit primes available for use are 2, 3, 5, and 7.
The problem states that both individual digits of the target number must be prime.
2
Determine valid units digits.
The units digit can only be 3 or 7.
If the units digit is 2, the number is even. If the units digit is 5, the number is a multiple of 5. Both cases result in a composite two-digit number.
3
List all potential combinations.
The possible combinations are 23, 33, 53, 73, 27, 37, 57, and 77.
These are generated by pairing any prime tens digit {2, 3, 5, 7} with the valid prime units digits {3, 7}.
4
Eliminate composite numbers from the list.
The valid primes are 23, 37, 53, and 73. The numbers 27, 33, 57, and 77 are removed.
27, 33, and 57 are divisible by 3 (sum of digits is a multiple of 3). 77 is divisible by 7.
5
Calculate the final sum.
23 + 73 = 96.
The question asks for the sum of the smallest valid number (23) and the largest valid number (73).

Key Concept

Classification of prime digits and prime numbers.
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Topics

Classification of Numbers

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