A fruit vendor has 260 mangoes , 292 oranges and 220 apples. He cells each of these fruits in a package containing the same number of fruits in a package containing the same number of fruits of the same kind and has found that 4 apples, 4 orangles and 4 mangoes are left out.
Find the minimum number of fruit packages.

Answer

To solve the problem step by step, we need to determine how many fruit packages the vendor can create from the given quantities of fruits, while accounting for the leftovers.

**Step 1: Determine the effective number of fruits after accounting for leftovers.**
- The vendor has:
- Mangoes: 260
- Oranges: 292
- Apples: 220
- Since 4 of each fruit are left out, we subtract these from the total:
- Effective mangoes = 260 - 4 = 256
- Effective oranges = 292 - 4 = 288
- Effective apples = 220 - 4 = 216

**Step 2: Find the greatest common divisor (GCD) of the effective quantities.**
- We need to find the GCD of 256, 288, and 216 to determine the size of each package.

**Finding GCD:**
- Prime factorization:
- 256 = 2^8
- 288 = 2^5 × 3^2
- 216 = 2^3 × 3^3

- The GCD is found by taking the lowest power of each prime factor:
- For 2: min(8, 5, 3) = 3 → 2^3
- For 3: min(0, 2, 3) = 0 → 3^0 (not included)

- Therefore, GCD = 2^3 = 8.

**Step 3: Calculate the number of packages for each fruit.**
- Now that we know each package will contain 8 fruits:
- Number of mango packages = Effective mangoes / Package size = 256 / 8 = 32
- Number of orange packages = Effective oranges / Package size = 288 / 8 = 36
- Number of apple packages = Effective apples / Package size = 216 / 8 = 27

**Step 4: Calculate the total number of packages.**
- Total packages = Number of mango packages + Number of orange packages + Number of apple packages
- Total packages = 32 + 36 + 27 = 95

**Final Answer:**
The minimum number of fruit packages is **95**.

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