Lekhana's rice bags

Lekhana buys rice from two farms and sells it in the market. She bought 84 kg from one farm and 108 kg from the other.

She wants the rice packed into bags, with each bag holding rice from only one farm, and all bags the same weight — a whole number of kilograms. If she wants to use as few bags as possible, what should each bag weigh?

The shape of the problem is the same as Sameeksha's, though the words are different. To divide 84 kg into bags of equal weight with nothing left over, the bag weight must be a FACTOR of 84. To do the same with 108 kg, it must be a factor of 108.

The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.

Since she wants the SAME bag weight for both farms, the weight must be a common factor. The common factors of 84 and 108 are 1, 2, 3, 4, 6 and 12.

Any of those would work. But she wants to minimise the NUMBER of bags, and a heavier bag holds more rice, so fewer are needed. She should choose the largest: 12 kg — the HCF of 84 and 108.

You met this idea before as a game. In 'Jump Jackpot' from Grade 6, Grumpy puts treasure on two numbers and Jumpy picks a jump size, starting from 0, that must land on both. The longest jump size that works is exactly the HCF — because a jump size that lands on a number must be a factor of it.