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# help

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If all multiples of 3 and all multiples of 4 are removed from the list of whole numbers 1 through 100, then how many whole numbers are left?

Oct 6, 2018

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$$\text{let }U \text{ be the set of whole numbers }1-100 \\ \text{and let }V \text{ be }U \text{ with the multiples of 3 and 4 removed}\\ |V| = |U| - |\text{multiples of 3 < 100}| - |\text{multiples of }4 \leq 100| + |\text{multiples of both 3 and 4} \leq 100|$$

$$|U| = 100 \\ |\text{multiples of 3}<100| = 33 \\ |\text{multiples of 4}\leq 100| = 25 \\ |\text{multiples of 3 and 4}\leq 100| = |\text{multiples of 12}\leq 100| = 8 \\ |V| = 100 - 33 - 25 + 8 = 50$$

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Oct 6, 2018