Anshu and Guna's torans

Anshu and Guna make torans out of strips of cloth, placed one next to another. Anshu uses strips 6 cm long and Guna uses strips 8 cm long. If both must make torans of the SAME length, what is the smallest that length could be?

Any toran Anshu makes is built of 6 cm strips, so its length is a MULTIPLE of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, and so on.

Any toran Guna makes is a multiple of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, and so on.

For both to make a toran of the same length, that length must be a common multiple of 6 and 8. From the two lists, 24 and 48 are both common multiples — so torans of 24 cm and 48 cm are possible for both.

The smallest of them is 24. So 24 cm is the shortest toran both can stitch. (And what about the LARGEST common multiple? There is none: the lists go on for ever, so common multiples never run out.)

Here is the same idea again. A sweet shop gives out free gajak to school children every Monday — that is, every 7 days. Kabamai visits the shop once every 10 days, and today is a Monday when she got some. When will she next get free gajak?

The shop gives sweets after 7, 14, 21, 28, … days, the multiples of 7. Kabamai arrives after 10, 20, 30, 40, … days, the multiples of 10. The next free gajak is at the first day common to both lists: 70 days. Again the answer is the lowest common multiple. The Lowest Common Multiple (LCM) of two or more numbers (in Hindi, लघुत्तम समापवर्त्य) is the smallest of their common multiples — and it is the answer whenever two repeating cycles must coincide, as in the Idli-Vada game of Grade 6.