1. **State the problem:** We need to show that 43 is a prime number and find the prime factorization of 2560. 2. **Show that 43 is prime:** A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. 3. **Check divisibility of 43:** We test divisibility by prime numbers less than or equal to $\sqrt{43}$. 4. Calculate $\sqrt{43} \approx 6.56$, so test primes 2, 3, and 5. 5. 43 is not divisible by 2 (it's odd), not divisible by 3 (sum of digits 4+3=7, not multiple of 3), and not divisible by 5 (does not end with 0 or 5). 6. Since 43 has no divisors other than 1 and 43, it is prime. 7. **Prime factorization of 2560:** Start dividing by smallest primes. 8. Divide by 2: $2560 \div 2 = 1280$ 9. Divide by 2: $1280 \div 2 = 640$ 10. Divide by 2: $640 \div 2 = 320$ 11. Divide by 2: $320 \div 2 = 160$ 12. Divide by 2: $160 \div 2 = 80$ 13. Divide by 2: $80 \div 2 = 40$ 14. Divide by 2: $40 \div 2 = 20$ 15. Divide by 2: $20 \div 2 = 10$ 16. Divide by 2: $10 \div 2 = 5$ 17. 5 is prime, so stop. 18. Count the number of 2's: 9 times. 19. Therefore, prime factorization of 2560 is $$2560 = 2^{9} \times 5$$. **Final answers:** - 43 is a prime number. - Prime factorization of 2560 is $$2^{9} \times 5$$.