{Next week|On Saturday}, `person(1)` is having a party{ and `he(1)`'s planning to play `his(1)`
`randRange(2,30)` favorite songs. `He(1)` also|. `He(1)`} wants to get some hot dogs for the party. When `he(1)`
goes to the store, `he(1)` finds that hot dogs come in packages of `A` and buns come in packages of `B`.

If `person(1)` wants to have the same number of hot dogs and buns,
what is the minimum number of hot dogs `he(1)` will have to buy?

`LCM`

We know that hot dogs come in packages of

. Write out the first few multiples
of `A`

to see the possible numbers of hot dogs `A``person(1)` can buy:

`\qquad dogs: `

`M`,`...`

We know that buns come in packages of

. Write out the first few multiples
of `B`

to see the possible numbers of buns `B``person(1)` can buy:

`\qquad buns: `

`M`,`...`

Since `person(1)` wants to have the same number of hot dogs and buns, look for *common multiples* where it's
possible to buy the same number of hot dogs and buns:

`\qquad dogs: `

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

`\qquad buns: `

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

The *least* common multiple is the minimum number of hot dogs `person(1)` will have to buy to get
the same number of hot dogs and buns.

is the least common multiple of `LCM`

and `A`

.`B`

To get the same number of each, **the smallest amount of food person(1) can buy is LCM hot dogs and buns**,
or

`person(1)` is organizing a {baseball|softball} league, and `he(1)` needs to purchase jerseys and
visors for the players. Jerseys come in sets of `A`, and visors come in sets of `B`.

If `person(1)` wants to buy the same number of jerseys and visors,
what is the minimum number of jerseys or visors `he(1)` will have to purchase?

`LCM`

We know that jerseys come in packages of

. Write out the first few multiples
of `A`

to see the possible numbers of jerseys `A``person(1)` can buy:

`\qquad jerseys: `

`M`,`...`

We know that visors come in packages of

. Write out the first few multiples
of `B`

to see the possible numbers of visors `B``person(1)` can buy:

`\qquad visors: `

`M`,`...`

Since `person(1)` wants to have the same number of jerseys and visors, look for *common multiples* where it's
possible to buy the same number of jerseys and visors:

`\qquad jerseys: `

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

`\qquad visors: `

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

The *least* common multiple is the minimum number of jerseys `person(1)` will have to buy to get
the same number of jerseys and visors.

is the least common multiple of `LCM`

and `A`

.`B`

To get the same number of each, **the smallest number person(1) can buy is LCM jerseys and visors**,
or

`person(1)` and `person(2)` are in different `course(1)` classes at `school(1)`.
`person(1)`'s teacher always gives `plural( exam(1) )` with `A` questions on them while
`person(2)`'s teacher gives more frequent `plural( exam(1) )` with only `B` questions.
`person(2)`'s teacher always gives `plural( exam(1) )` with `B` questions on them while
`person(1)`'s teacher gives more frequent `plural( exam(1) )` with only `A` questions.
{`person(1)` has `randRange(15,40)` other students in `his(1)` class.
|`person(2)`'s teacher also assigns `randRange(3,10)` projects per year.}

Even though the two classes have to take a different number of `plural( exam(1) )`, their teachers have
told them that both classes will get the same total number of `exam(1)` questions each year.

What is the minimum number of `exam(1)` questions `person(1)`'s or `person(2)`'s class
can expect to get in a year?

`LCM`

We know that in `person(1)`'s class, all the `plural( exam(1) )` have

questions. Write
out the first few multiples of `A`

to see the possible numbers of questions
`A``person(1)` might have to answer over the whole year:

`\qquad`

`M`,`...`

We know that in `person(2)`'s class, all the `plural( exam(1) )` have

questions. Write
out the first few multiples of `B`

to see the possible numbers of questions
`B``person(2)` might have to answer over the whole year:

`\qquad`

`M`,`...`

Since `person(1)`'s and `person(2)`'s teachers have told them that both classes will have the same total
number of `exam(1)` questions over the whole year, look for the *common multiples* to find the possible
numbers of `exam(1)` questions they will have to answer.

`\qquad`

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

`\qquad`

`\color{`

`PINK`}{`M`},`\color{`

`BLUE`}{`M`},`\color{`

`GREEN`}{`M`},`M`,`...`

The *least* common multiple is the minimum number questions `person(1)` and `person(2)` might
have to answer over the year.

is the least common multiple of `LCM`

and `A`

.`B`

If `person(1)`'s and `person(2)`'s classes get the same total number of questions,
**the minimum number of exam(1) questions they can expect to get in a year is LCM questions**,
or

At a track and field competition, there are `A` sprinters and `B` long-distance
runners{ and `randRange(5,100)` fans|}. `person(1)` has to assign all of the athletes
to teams. `He(1)` wants to make sure all of the teams have the same number of sprinters and
the same number of long-distance runners.

**What is the greatest number of teams person(1) can form?**

`GCD`

Let's start by just thinking about the sprinters. We can think about all the ways to divide the
`A` sprinters into equally sized teams by finding the factors of `A`.

The factors of `A` are `toSentence( getFactors( A ) )` since
those are all the numbers that divide evenly into `A`.
That means we can divide the sprinters into equally sized teams in any of the following ways:

`plural( F, "team" )` with `plural( A / F, "sprinter" )` `plural( "", "each", F )`

Now lets think about the long-distance runners. We can also list all the ways to divide the
`B` long-distance runners into equally sized teams by finding the factors of `B`.

The factors of `B` are `toSentence( getFactors( B ) )` since
those are all the numbers that divide evenly into `B`.
That means we can divide the long-distance runners into equally sized teams in any of the following ways:

`plural( F, "team" )` with `plural( B / F, "long-distance runner" )` `plural( "", "each", F )`

Since each team will have sprinters and long-distance runners, compare the numbers of teams the sprinters can be divided into and
the numbers of teams the runners can be divided into to find the *common divisors*:

The common divisors of `A` and `B` are `toSentence( _.intersection( A_FACTORS, B_FACTORS ) )`.
In other words, with `A` sprinters and `B` long-distance runners,
`person(1)` can make the following equal teams:

`plural( F, "team" )` with `plural( A / F, "sprinter" )` and
`plural( B / F, "long-distance runner" )`

We want to know the *greatest* number of equal teams `person(1)` can make, so from the common divisors above,
we want the *greatest common divisor*.

**The greatest number of teams that person(1) can form is GCD teams**,
with

At `person(1)`'s bakery, `person(1)` bakes one batch of `A` chocolate chip cookies
and one batch of `B` oatmeal cookies each day. `person(1)` sells all `his(1)` cookies
the same day in gift baskets.

{Because `his(1)` customers expect consistency|To keep the price the same}, `person(1)` wants
to make sure each gift basket is identical.

What is the greatest number of gift baskets `person(1)` can sell each day?

`GCD`

Let's start by just thinking about the chocolate chip cookies. We can think about all the ways to equally
divide the `A` chocolate chip cookies into gift baskets by finding the factors of `A`.

The factors of `A` are `toSentence( getFactors( A ) )` since
those are all the numbers that divide evenly into `A`.
That means we can equally divide the chocolate chip cookies into gift baskets in any of the following ways:

`plural( F, "basket" )` with `plural( A / F, "chocolate chip cookie" )` `plural( "", "each", F )`

Now lets think about the oatmeal cookies. We can also list all the ways to equally divide the
`B` oatmeal cookies into gift baskets by finding the factors of `B`.

The factors of `B` are `toSentence( getFactors( B ) )` since
those are all the numbers that divide evenly into `B`.
That means we can equally divide the oatmeal cookies into gift baskets in any of the following ways:

`plural( F, "basket" )` with `plural( B / F, "oatmeal cookie" )` `plural( "", "each", F )`

Since each gift basket will have chocolate chip and oatmeal cookies, compare the ways of dividing the chocolate chip cookies
and the ways of dividing the oatmeal cookies to find the *common divisors*:

The common divisors of `A` and `B` are `toSentence( _.intersection( A_FACTORS, B_FACTORS ) )`.
In other words, with `A` chocolate chip and `B` oatmeal cookies,
`person(1)` can make any of the following gift baskets:

`plural( F, "basket" )` with `plural( A / F, "chocolate chip cookie" )` and
`plural( B / F, "oatmeal cookie" )`

We want to know the *greatest* number of identical gift baskets `person(1)` can make, so from the common divisors above,
we want the *greatest common divisor*.

**The greatest number of gift baskets that person(1) can make each day is GCD baskets**,
with

`person(1)` just bought 1 package of `plural( A, deskItem(1) )` and 1 package of
`plural( B, deskItem(2) )`. `He(1)` wants to use all of the `plural( deskItem(1) )`
and `plural( deskItem(2) )` to create identical sets of office supplies for `his(1)`
{coworkers|friends|classmates}.

What is the greatest number of sets of office supplies `person(1)` can make?

`GCD`

Let's start by just thinking about the `plural( deskItem(1) )`. We can think about all the ways to
equally divide the `plural( A, deskItem(1) )` into sets by finding the factors of `A`.

The factors of `A` are `toSentence( getFactors( A ) )` since
those are all the numbers that divide evenly into `A`.
That means we can equally divide the `plural( deskItem(1) )` into sets in any of the following ways:

`plural( F, "set" )` with `plural( A / F, deskItem(1) )` `plural( "", "each", F )`

Now lets think about the `plural( deskItem(2) )`. We can also list all the ways to equally divide the
`plural( B, deskItem(2) )` into sets by finding the factors of `B`.

The factors of `B` are `toSentence( getFactors( B ) )` since
those are all the numbers that divide evenly into `B`.
That means we can equally divide the `plural( deskItem(2) )` into sets in any of the following ways:

`plural( F, "set" )` with `plural( B / F, deskItem(2) )` `plural( "", "each", F )`

Since each set will have `plural( deskItem(1) )` and `plural( deskItem(2) )`, compare the
ways of dividing the `plural( deskItem(1) )` and the ways of dividing the
`plural( deskItem(2) )` to find the *common divisors*:

The common divisors of `A` and `B` are `toSentence( _.intersection( A_FACTORS, B_FACTORS ) )`.
In other words, with `plural( A, deskItem(1) )` and `plural( B, deskItem(2) )`,
`person(1)` can make any of the following sets:

`plural( F, "set" )` with `plural( A / F, deskItem(1) )` and
`plural( B / F, deskItem(2) )`

We want to know the *greatest* number of identical sets `person(1)` can make, so from the common divisors above,
we want the *greatest common divisor*.

**The greatest number of sets of office supplies that person(1) can make is GCD sets**,
with