The school cafeteria serves pizza every sixth day and applesauce every eighth day. If pizza and applesauce are both on today's menu, in how many days will they be together on the menu again?

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We need to find the least common multiple of 6 and 8.

First factor both numbers as a product of primes:

`6=2*3; 8=2*2*2=2^3 `

The least common multiple can be found by taking the product of each prime factor that occurs to the highest power that it occurs in either...

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We need to find the least common multiple of 6 and 8.

First factor both numbers as a product of primes:

`6=2*3; 8=2*2*2=2^3 `

The least common multiple can be found by taking the product of each prime factor that occurs to the highest power that it occurs in either factorization.

We have a factor of 2 which occurs to the 3rd power, and a factor of 3.

Thus LCM(6,8)=`2^3*3=24 `

The next day the two occur together is 24 days later.

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To solve, let's apply the least common multiple (LCM) of a set of numbers.

Pizza is served every sixth day. To get the days in which it will be served, take the multiples of 6.

The multiples of 6 are:

6, 12, 18, 24, 30, ... 6n

So pizza will be served on day 6, day 12, day 18, day 24, day 30 until day 6n.

Applesauce is served every 8th day. To get the days in which it will be served, take the multiples of 8.

8, 16, 24, 32, 40, ... 8n

So applesauce will be served on day 8, day 16, day 24, day 32, day 40 until day 8n.

Base on the list above, the LCM of 6 and 8 is 24.

Therefore, the pizza and apple sauce are served together every 24th day.

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