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# Overthinking

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Let A be the multiples of 4 between {0-170) and B be the multiples of 6 between {0-170}. How many numbers are in the intersection of A and B.

Isn't the intersection the LCM of 4 and 6, which is 12. so, 170/12?

Dec 2, 2018

#1
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Not sure, I'm getting 56?

Dec 2, 2018
#2
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Multiples of 6 and 4 from 0 - 170 =

12, 24, 36, 48, 60,  72,  84, 96, 108, 120,132,  144, 156, 168   Dec 2, 2018
#3
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Yes! Thank you! It is 14, right?

ant101  Dec 2, 2018
#4
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Correct, Ant !!!   CPhill  Dec 2, 2018
#5
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Hello! Here is a long way!

If you know the principle of Inclusion and Exclusion, this will lead to the size of their union, but in this case, we have to find the intersection! Therefore, we can find the size of union and subtract from the number of multiples of 4 and 6. There are $$\frac{170}{4}\approx42$$ multiples of $$4$$ and $$\frac{170}{6}\approx28$$ multiples of $$6.$$The least common multiple of 4 and 6 is 12, so $$\frac{170}{12}\approx14$$ . You could stop here, because this is your answer, but let's continue and show why this works. The union is just $$42+28-14=56$$ and there are $$42+28$$ total multiple of 4 and 6. Thus, the intersection is $$70-56=\boxed{14}$$ . You could've stopped well before and found your answer, but here is just a long way.

Dec 2, 2018
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Thanks, tertre.....!!!!   CPhill  Dec 2, 2018