1. **Problem Statement:** Shalvi has 36 apples and 60 bananas and wants to distribute them equally among guests without discrimination. 2. **Formula and Concept:** To find the maximum number of guests, we need to find the greatest common divisor (GCD) of the number of apples and bananas. The GCD gives the largest number that divides both quantities exactly. 3. **Calculate GCD of 36 and 60:** $$\text{GCD}(36, 60)$$ Prime factors: - 36 = $2^2 \times 3^2$ - 60 = $2^2 \times 3 \times 5$ Common factors: $2^2 \times 3 = 12$ 4. **Maximum guests:** $$\text{Maximum guests} = 12$$ 5. **Fruits per guest:** - Apples per guest = $\frac{36}{12} = 3$ - Bananas per guest = $\frac{60}{12} = 5$ 6. **Adding 42 mangoes:** Now Shalvi has 36 apples, 60 bananas, and 42 mangoes. 7. **Find GCD of 36, 60, and 42:** Prime factors: - 36 = $2^2 \times 3^2$ - 60 = $2^2 \times 3 \times 5$ - 42 = $2 \times 3 \times 7$ Common factors: $2^\cancel{2} \times 3 = 6$ 8. **Maximum guests with mangoes:** $$\text{Maximum guests} = 6$$ 9. **Fruits per guest now:** - Apples per guest = $\frac{36}{6} = 6$ - Bananas per guest = $\frac{60}{6} = 10$ - Mangoes per guest = $\frac{42}{6} = 7$ 10. **Total fruits per guest:** $$6 + 10 + 7 = 23$$ **Final answers:** - (i) Maximum guests without mangoes: 12 - (ii) Each guest gets 3 apples and 5 bananas - (iii) With mangoes, maximum guests: 6 - Total fruits per guest with mangoes: 23