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Question Number 2381 by Yozzi last updated on 18/Nov/15

$$Ofthenumbers1,2,3,...,6000,$$$$howmanyarenotmultiplesof2,3or5?$$

Commented by 123456 last updated on 18/Nov/15

$$\mathrm{N}={\mathrm{N}}_2+{\mathrm{N}}_3+{\mathrm{N}}_5-{\mathrm{N}}_{2,3}-{\mathrm{N}}_{2,5}-{\mathrm{N}}_{3,5}+{\mathrm{N}}_{2,3,5}$$$$\mathrm{M}=\mathrm{X}-\mathrm{N}$$

Commented by prakash jain last updated on 18/Nov/15

$$n(\mathrm{A}\cup \mathrm{B}\cup \mathrm{C})=n\left(\mathrm{A}\right)+n\left(\mathrm{B}\right)+n\left(\mathrm{C}\right)$$$$-n(\mathrm{A}\cap \mathrm{B})-n(\mathrm{A}\cap \mathrm{C})-n(\mathrm{B}\cap \mathrm{C})$$$$+n(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C})$$

Answered by 123456 last updated on 18/Nov/15

$$\mathrm{first}\mathrm{we}\mathrm{have}\mathrm{that}1,2,...,6000\mathrm{have}$$$$6000\mathrm{numer}(\mathrm{X}=6000)$$$$\mathrm{lets}\mathrm{count}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{multiples}\mathrm{of}$$$$2,3,5,\mathrm{for}\mathrm{this},\mathrm{we}\mathrm{only}\mathrm{need}\mathrm{to}\mathrm{count}$$$$\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{multiply}\mathrm{of}\mathrm{these}\mathrm{and}$$$$\mathrm{subtract}\mathrm{tese}\mathrm{that}\mathrm{are}\mathrm{multply}\mathrm{of}\mathrm{two}$$$$\mathrm{and}\mathrm{sum}\mathrm{the}\mathrm{multiple}\mathrm{of}\mathrm{tree}\mathrm{of}\mathrm{these}$$$$\mathrm{ps}:-\lceil \frac{1}{a}\rceil +1=0,a\in {\mathbb{N}}^{\ast }$$$${\mathrm{N}}_2=\lfloor \frac{6000}{2}\rfloor =3000$$$${\mathrm{N}}_3=\lfloor \frac{6000}{3}\rfloor =2000$$$${\mathrm{N}}_5=\lfloor \frac{6000}{5}\rfloor =1200$$$${\mathrm{N}}_{2,3}=\lfloor \frac{6000}{2\times 3}\rfloor =\lfloor \frac{6000}{6}\rfloor =1000$$$${\mathrm{N}}_{2,5}=\lfloor \frac{6000}{2\times 5}\rfloor =\lfloor \frac{6000}{10}\rfloor =600$$$${\mathrm{N}}_{3,5}=\lfloor \frac{6000}{3\times 5}\rfloor =\lfloor \frac{6000}{15}\rfloor =400$$$${\mathrm{N}}_{2,3,5}=\lfloor \frac{6000}{2\times 3\times 5}\rfloor =\lfloor \frac{6000}{30}\rfloor =200$$$$\mathrm{N}=3000+2000+1200-1000-600-400+200$$$$\mathrm{N}=4400$$$$\mathrm{so},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{no}\mathrm{multiple}\mathrm{is}\mathrm{the}$$$$\mathrm{total}\mathrm{minus}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{multiples}$$$$\mathrm{M}=6000-4400=1600$$

Commented by Yozzi last updated on 18/Nov/15

$$Thanks.Igotthat.$$

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