There are 75 houses in the troll's street. if in every 4th house there is a pink troll, a blue troll in every 6th house and a yellow troll in every 8th house, then in how many houses are there all of pink, blue and yellow trolls?

3

Explanation:

The problem is first to find the Least Common Multiple (LCM) for the numbers 4, 6 and 8 together.

To find the LCM, we have to break down every number into it's prime factors:

4: 2 x 2 = 2²

6: 2 x 3 = 2 x 3

8: 2 x 2 x 2 = 2³

Then for each prime number listed we pick the highest exponent and we multiply them.  For 2, the highest comes from 8 (2³), while for 3 there's only one occurrence... so the choice is easy:

LCM (4,6,8) = 2³ * 3 = 8 * 3 = 24

So, every 24 houses, we'll find all three trolls colors.

How many times will that happen within a group of 75 houses?

3 times of course, at houses #24, #48 and #72.

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