ShmoopTube

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Word Problems with Multiples, a la Shmoop.

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Welcome one, welcome all, to the city of AV. [Welcome sign for city of AV]

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That’s…short for AcronymVille.

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They abbreviate everything here. If you need to mail a letter, you go to the P.O.

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If you’re sick, you drink some C.N.S. [Sick woman drinking chicken noodle soup]

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If you had too many beans for lunch, you might have to make a run for the WC.

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You get the picture.

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The two big cheeses in this town are the joint mayors – LCM and GCF. [Joint mayors in an office]

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LCM stands for Least Common Multiple, and GCF stands for Greatest Common Factor.

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These two are always disagreeing about the right way to run AV. [LCM and GCF fighting with boxing gloves]

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For example, a few years ago when the two were discussing the growing W.B. problem –

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– that’s W.B. for “Water Buffalo”…

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…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]

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48 male W.B.’s and 36 female W.B.’s had been sighted in the area. GCF felt that each

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pen should have the same number of males and the same number of females, and that as few

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W.B.’s should be kept in each pen as possible.

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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]

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Finding the prime factors of each number, GCF found that 48 broke down into 2 times

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2 times 2 times 2 times 3…

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…while 36 broke down into 2 times 2 times 3 times 3.

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The overlap between the two was 2 times 2 times 3…which comes out to 12.

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Meaning…that the maximum number of pens that could be built that would keep the same

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numbers of males and females in each pen… would be 12. [prime factors of the two pens]

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To see how many males and females are kept in each pen, just divide the original totals,

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48 and 36, by the number of pens, 12.

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4 males to 3 females in each. But after GCF put his plan in action, LCM

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decided it would be better to make the W.B.s the city’s major export… [Truck picks up water buffalo and rides away]

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and sell them to neighboring towns.

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Since each pen now housed 4 males and 3 females, he wanted to find the Least Common Factor

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of sets of each gender, so that he could put together packages of an equal number of males

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or females to attract potential buyers.

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It turns out that water buffaloes can be very clique-y. [Water buffaloes drinking water]

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For example, he knew he could pull 3 females out of one pen and 3 out of another…

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…but there would be no way to nab 6 males without upsetting the delicate balance.

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To find the number he was looking for, LCM made this Venn Diagram, that included the [Venn diagram of prime factors of the pens]

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prime factors of both 4 and 3…found where they overlapped, then multiplied all those

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numbers together.

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Since 2 times 2 times 1 times 3 is 12…LCM knew he could take 12 males – 3 sets of

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4 – and 12 females – 4 sets of 3, without throwing things out of whack.

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The two still argue to this day about how the situation ideally should have been handled, [GCF and LCM arguing in a meeting room]

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but deep down we think they consider one another good friends.

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Er, sorry…BFF’s. [GCF and LCM hugging each other]

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