The sum of three prime numbers is 100. One of these prime numbers exceeds another by 36. What is one of the three prime numbers?

Introduction / Context:
This question tests your understanding of prime numbers and your ability to use simple algebra to model numerical relationships. It also checks reasoning with constraints on sums and differences of primes.

Given Data / Assumptions:

• We have three prime numbers p, q and r.
• Their sum is p + q + r = 100.
• One of them exceeds another by 36, so we can set p = q + 36 without loss of generality.
• All three numbers are prime.

Concept / Approach:
We express the conditions using algebra: p = q + 36 and p + q + r = 100. Substituting p in terms of q gives a relation between q and r. Then we search for prime pairs that satisfy this relation. We also remember that the only even prime is 2, and the sum of three odd primes is odd, which gives a parity constraint.

Step-by-Step Solution:
Step 1: Assume p and q are the primes where p = q + 36.Step 2: Use the sum condition: p + q + r = 100 ⇒ (q + 36) + q + r = 100 ⇒ 2q + r = 64.Step 3: Since the sum of three primes is 100, and 100 is even, either all three primes are odd with one of them equal to 2, or two are odd and one is even (2). But the sum of three odd primes would be odd, so one of the primes must be 2.Step 4: Try r = 2. Then 2q + 2 = 64 ⇒ 2q = 62 ⇒ q = 31.Step 5: Then p = q + 36 = 31 + 36 = 67.Step 6: Check primality: 31, 67 and 2 are all prime.Step 7: So the three primes are 2, 31 and 67. None of the options 17, 29 or 43 appears in this set.

Verification / Alternative check:
Confirm the sum: 2 + 31 + 67 = 100.Confirm the difference condition: 67 - 31 = 36, so one prime does indeed exceed another by 36.Therefore the triple (2, 31, 67) satisfies all conditions, and the given options do not list any of these primes.

Why Other Options Are Wrong:
17, 29 and 43 are all prime numbers individually, but none of them belongs to the triple 2, 31 and 67 that satisfies both the sum and difference conditions.Choosing any of them would correspond to a different triple that does not meet the given constraints.

Common Pitfalls:
Forgetting that the only even prime is 2 can lead to incorrect assumptions about the parity of the primes.Students may try random triplets without using the algebraic relation 2q + r = 64, which wastes time and increases the chance of mistakes.Systematic reasoning and the parity argument make the solution much cleaner.

Final Answer:
None of the given numbers is one of the three primes; the correct prime triple is 2, 31 and 67.