Consider the numbers 8, 13, 29 and 38. Three of these numbers can be expressed as the sum of two prime numbers, while one number cannot. Which number does not belong to that group?

Introduction / Context:
This question is a number classification problem that quietly uses knowledge about prime numbers and their sums. The test checks whether you can identify a property related to representation of numbers as sums of two primes and then use that property to classify the given numbers. One number will break the pattern and will therefore be the odd one out.

Given Data / Assumptions:
The numbers presented are 8, 13, 29 and 38.
You should recall what a prime number is – a natural number greater than 1 that has no positive divisors other than 1 and itself.
The question suggests that the classification may rely on a structural property, such as representation of each number as a sum of two prime numbers.
Exactly one of these numbers will fail to have the same property as the others.

Concept / Approach:
A useful idea here is inspired by the general concept that many numbers can be written as sums of two primes. If you can show that three of the given numbers can be expressed as a sum of two prime numbers, while one number cannot be so expressed, then the number that fails will be the odd one out. This gives a clear and logical distinction that separates one number from the rest.

Step-by-Step Solution:
Step 1: Start with 8. Try writing 8 as a sum of two primes. We have 3 + 5 = 8, and both 3 and 5 are prime numbers. So 8 can be expressed as a sum of two primes. Step 2: Consider 13. Try combinations like 2 + 11 = 13. Here, 2 and 11 are both prime numbers, so 13 can also be expressed as a sum of two primes. Step 3: Consider 38. We can write 38 as 19 + 19. Both 19 and 19 are prime numbers, so 38 is also expressible as a sum of two primes. Step 4: Now examine 29. Try possible pairs of primes that could sum to 29. Start with 2: 29 - 2 = 27, but 27 is not prime. Next 3: 29 - 3 = 26, not prime. Next 5: 29 - 5 = 24, not prime. Next 7: 29 - 7 = 22, not prime. Next 11: 29 - 11 = 18, not prime. Next 13: 29 - 13 = 16, not prime. We run out of options without finding two primes that sum to 29. Step 5: Therefore, 29 cannot be represented as the sum of two prime numbers, while 8, 13 and 38 can. This makes 29 different from the others.

Verification / Alternative check:
To be thorough, you can check that there is no overlooked pair of primes. Since one of the primes must be at most half of 29, you only need to test primes 2, 3, 5, 7, 11 and 13 as one of the addends. None of the corresponding differences are prime, so there is no valid prime pair for 29. For the other numbers, we already have explicit decompositions into sums of primes. This confirms that 29 alone fails the property.

Why Other Options Are Wrong:
8: Can be written as 3 + 5, both primes, so it satisfies the required property.
13: Can be written as 2 + 11, again both prime numbers, so it fits the pattern.
38: Can be written as 19 + 19, with both addends prime, so it also belongs to the main group.

Common Pitfalls:
Students sometimes look only at simple properties like “odd versus even” or “prime versus composite”. Here 13 and 29 are prime, while 8 and 38 are composite, which does not isolate a single odd one out. The question instead uses a slightly more subtle property. Remember that in classification problems with numbers, it often helps to think about how numbers can be decomposed using primes, squares, or other structures, not just their basic parity or divisibility.

Final Answer:
The unique number that cannot be expressed as a sum of two prime numbers is 29.