Unit 3.7 Word Problems with Multiples
If you had too many beans for lunch, you might have to make a run for the WC.
You get the picture.
The two big cheeses in this town are the joint mayors – LCM and GCF. [Joint mayors in an office]
LCM stands for Least Common Multiple, and GCF stands for Greatest Common Factor.
These two are always disagreeing about the right way to run AV. [LCM and GCF fighting with boxing gloves]
For example, a few years ago when the two were discussing the growing W.B. problem –
– that’s W.B. for “Water Buffalo”…
…GCF suggested they capture all the W.B.’s that were roaming the streets and move them into pens. [Hand picks up water buffalo's]
48 male W.B.’s and 36 female W.B.’s had been sighted in the area. GCF felt that each
pen should have the same number of males and the same number of females, and that as few
W.B.’s should be kept in each pen as possible.
Don’t ask where he came up with that. He is just an elected official. He’s got some weird ideas. [GCF giving a speech]
Finding the prime factors of each number, GCF found that 48 broke down into 2 times
2 times 2 times 2 times 3…
…while 36 broke down into 2 times 2 times 3 times 3.
The overlap between the two was 2 times 2 times 3…which comes out to 12.
Meaning…that the maximum number of pens that could be built that would keep the same
numbers of males and females in each pen… would be 12. [prime factors of the two pens]
To see how many males and females are kept in each pen, just divide the original totals,
48 and 36, by the number of pens, 12.
4 males to 3 females in each. But after GCF put his plan in action, LCM
decided it would be better to make the W.B.s the city’s major export… [Truck picks up water buffalo and rides away]
and sell them to neighboring towns.
Since each pen now housed 4 males and 3 females, he wanted to find the Least Common Factor
of sets of each gender, so that he could put together packages of an equal number of males
or females to attract potential buyers.
It turns out that water buffaloes can be very clique-y. [Water buffaloes drinking water]
For example, he knew he could pull 3 females out of one pen and 3 out of another…
…but there would be no way to nab 6 males without upsetting the delicate balance.
To find the number he was looking for, LCM made this Venn Diagram, that included the [Venn diagram of prime factors of the pens]
prime factors of both 4 and 3…found where they overlapped, then multiplied all those
Since 2 times 2 times 1 times 3 is 12…LCM knew he could take 12 males – 3 sets of
4 – and 12 females – 4 sets of 3, without throwing things out of whack.
The two still argue to this day about how the situation ideally should have been handled, [GCF and LCM arguing in a meeting room]
but deep down we think they consider one another good friends.
Er, sorry…BFF’s. [GCF and LCM hugging each other]