1. **Problem:** There are between 50 and 60 eggs in a basket. When counted by 3's, remainder is 2. When counted by 5's, remainder is 4. Find the number of eggs. 2. **Formula and rules:** We use the concept of congruences: $$x \equiv 2 \pmod{3}$$ $$x \equiv 4 \pmod{5}$$ We look for a number $x$ between 50 and 60 satisfying both. 3. **Step-by-step solution:** - Numbers between 50 and 60: 51, 52, 53, 54, 55, 56, 57, 58, 59 - Check each for $x \equiv 2 \pmod{3}$: - 50 mod 3 = 2 (50 not in range) - 53 mod 3 = 2 - 56 mod 3 = 2 - Check these for $x \equiv 4 \pmod{5}$: - 53 mod 5 = 3 (no) - 56 mod 5 = 1 (no) - Check 59: - 59 mod 3 = 2 - 59 mod 5 = 4 4. **Answer:** The number of eggs is $\boxed{59}$.