Question: In 1742, the mathematician Christian Goldbach said that every even number greater than 2 can be written as the sum of two primes. Nobody knows if this is true or not. a) Find two primes that add together to make 30 b) Choose an even number greater than 50 and find two primes that add together to make your number.

1. **Problem Statement:** We need to find two prime numbers that add up to a given even number. 2. **Recall Goldbach's Conjecture:** It states every even number greater than 2 can be expressed as the sum of two primes. 3. **Part (a): Find two primes that add to 30** - List some primes less than 30: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ - Check pairs that sum to 30: - $7 + 23 = 30$ - $13 + 17 = 30$ - So, two valid pairs are $(7, 23)$ and $(13, 17)$. 4. **Part (b): Choose an even number greater than 50 and find two primes that add to it** - Let's choose $56$. - List primes less than 56: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53$ - Check pairs that sum to 56: - $3 + 53 = 56$ - $13 + 43 = 56$ - $19 + 37 = 56$ - So, valid pairs include $(3, 53)$, $(13, 43)$, and $(19, 37)$. **Final answers:** - (a) $7 + 23 = 30$ or $13 + 17 = 30$ - (b) For $56$, $3 + 53 = 56$ (among others)