A fruit seller has 405 mangoes, 675 oranges and 540 apples. She packs them in boxes so that each box holds only one kind of fruit and every box holds the same number of fruits. What is the least number of boxes she needs?

Explanation

1. 01

   Fewest boxes means most fruits per box: HCF(405, 675, 540).

2. 02

   $405 = 3^4 \times 5$, $675 = 3^3 \times 5^2$, $540 = 2^2 \times 3^3 \times 5$, so HCF = $3^3 \times 5 = 135$.

3. 03

   Boxes = $405 \div 135 + 675 \div 135 + 540 \div 135 = 3 + 5 + 4 = 12$.