Question

A three-digit number between 100 and 200 is prime. The number, formed by the last two digits of it is also prime. The two last digits themselves are also prime! What is the 3-digit number?

Answer 1

The answer is 113, 137, 139, 157, 179, and 197. These are the prime numbers between 100 and 200 where the last two digits also form a prime number, and each of the last two digits is individually a prime number.

Step-by-step explanation:

The question asks for a three-digit prime number between 100 and 200 where the last two digits also form a prime number, and each of the last two digits is individually a prime number as well. Knowing that prime numbers (other than 2 and 5) end in 1, 3, 7, or 9, we start by identifying prime numbers between 10 and 99, because only these can form the last two digits of our three-digit prime number. The prime numbers between 10 and 99 that end with a prime digit are 11, 13, 17, 19, 31, 37, 71, 73, 79, and 97. However, since our three-digit number must be between 100 and 200, only those starting with '1' need to be considered. Evaluating these, 113, 131, 137, 139, 151, 157, 159, 173, 179, and 197 are the possible candidates for the three-digit prime numbers satisfying all conditions. By checking the primality of each, we find that 113, 137, 139, 157, 179, and 197 remain as possible answers. Because all the given conditions must be satisfied, we can list 113, 137, 139, 157, 179, and 197 as the valid solutions to the question.

Answer 2

The three-digit number that meets all the given conditions (prime between 100 and 200, last two digits prime, and the last two digits themselves prime) is 137.

Let's break down the problem step by step to find the three-digit number that satisfies all the given conditions.

Condition 1: A three-digit number between 100 and 200 is prime.

This means we need to find prime numbers between 100 and 200. Prime numbers in this range are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, and 193.

Condition 2: The number formed by the last two digits of it is also prime.

Let's consider each of the prime numbers from the list and check if the number formed by its last two digits is also prime.

• 101: The last two digits are 01. 01 is not a prime number.
• 103: The last two digits are 03. 03 is not a prime number.
• 107: The last two digits are 07. 07 is a prime number.
• 109: The last two digits are 09. 09 is not a prime number.
• 113: The last two digits are 13. 13 is a prime number.
• 127: The last two digits are 27. 27 is not a prime number.
• 131: The last two digits are 31. 31 is a prime number.
• 137: The last two digits are 37. 37 is a prime number.
• 139: The last two digits are 39. 39 is not a prime number.
• 149: The last two digits are 49. 49 is not a prime number.
• 151: The last two digits are 51. 51 is not a prime number.
• 157: The last two digits are 57. 57 is not a prime number.
• 163: The last two digits are 63. 63 is not a prime number.
• 167: The last two digits are 67. 67 is a prime number.
• 173: The last two digits are 73. 73 is a prime number.
• 179: The last two digits are 79. 79 is a prime number.
• 181: The last two digits are 81. 81 is not a prime number.
• 191: The last two digits are 91. 91 is not a prime number.
• 193: The last two digits are 93. 93 is not a prime number.

Condition 3: The two last digits themselves are also prime.

From the previous step, we have identified the numbers where the last two digits are prime: 107, 113, 131, 137, 167, 173, 179.

Now, let's check if the last two digits of these numbers are also prime.

• 107: The last two digits are 07. 07 is a prime number.
• 113: The last two digits are 13. 13 is a prime number.
• 131: The last two digits are 31. 31 is a prime number.
• 137: The last two digits are 37. 37 is a prime number.
• 167: The last two digits are 67. 67 is a prime number.
• 173: The last two digits are 73. 73 is a prime number.
• 179: The last two digits are 79. 79 is a prime number.

Now, we have found the numbers that satisfy all three conditions: 107, 113, 131, 137, 167, 173, 179.

So, the three-digit number that satisfies all the given conditions is one of these: 107, 113, 131, 137, 167, 173, 179.