   SEARCH HOME Math Central Quandaries & Queries  Question from Aris, a parent: A teacher has more than 100 sweets. She thinks that if she give 6, 8 or 9 sweets to her students she will have no remaining in the case. What is the smallest number of sweets in the bag? Hi Aris.

We have to assume the teacher is distributing the sweets evenly (the same number to each student).

If she distributes 6 candies per student, she has none left, so the number of candies is a multiple of 6.

Similarly, the number of candies is a multiple of 8 and of 9.

So you are looking for the lowest common multiple of 6, 8 and 9 which is above 100.

One way to find it is to first find the lowest common multiple under 100 by examining the prime factors:

6 = 2 x 3
8 = 2 x 2 x 2
9 = 3 x 3

So the lowest common multiple has at least three factors of two and two factors of three:

LCM = 23 x 32 = 72.

Any common multiple of 6, 8 and 9 must be a multiple of 72.

Can you take it from here?

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.