1. **Problem statement:** A farmer has 36 cows, 48 sheep, and 60 chickens. He wants to divide them into groups with the maximum number of animals in each group such that each group has the same number of cows, sheep, and chickens. 2. **Formula and concept:** To find the maximum number of animals in each group, we need to find the greatest common divisor (GCD) of the numbers 36, 48, and 60. The GCD is the largest number that divides all these numbers exactly. 3. **Find the GCD:** - Prime factorization of 36: $$36 = 2^2 \times 3^2$$ - Prime factorization of 48: $$48 = 2^4 \times 3$$ - Prime factorization of 60: $$60 = 2^2 \times 3 \times 5$$ 4. **Identify common prime factors with the smallest powers:** - For 2: minimum power is $2$ (since 36 and 60 have $2^2$, 48 has $2^4$) - For 3: minimum power is $1$ (since 48 and 60 have $3^1$, 36 has $3^2$) - For 5: not common to all three 5. **Calculate GCD:** $$\text{GCD} = 2^2 \times 3^1 = 4 \times 3 = 12$$ 6. **Interpretation:** The maximum number of animals in each group is 12. 7. **Check number of groups:** - Number of groups for cows: $$\frac{36}{12} = 3$$ - Number of groups for sheep: $$\frac{48}{12} = 4$$ - Number of groups for chickens: $$\frac{60}{12} = 5$$ Each group will have 12 animals, divided equally among cows, sheep, and chickens according to these ratios. **Final answer:** Each group will have 12 animals.