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Question Number 119035 by ZiYangLee last updated on 21/Oct/20

Among the positive integers less than 1200, how many of them are relatively prime to 60?

$$\mathrm{Among}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{less}\:\mathrm{than}\:\mathrm{1200}, \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime} \\ $$$$\mathrm{to}\:\mathrm{60}? \\ $$

Answered by floor(10²Eta[1]) last updated on 21/Oct/20

60=2^2 .3.5 so we want to know how many numbers that doesn′t have 2, 3 or 5 as factors 1200−⌊((1200)/2)⌋=600 numbers that are not divisible by 2 from 1 to 1200 600−⌊((600)/3)⌋=400 numbers that don′t have factor 3 or 2 from 1 to 1200 400−⌊((400)/5)⌋=320 numbers that doesn′t have factors 2, 3 or 5 from 1 to 1200 Answer: 320

$$\mathrm{60}=\mathrm{2}^{\mathrm{2}} .\mathrm{3}.\mathrm{5} \\ $$$$\mathrm{so}\:\mathrm{we}\:\mathrm{want}\:\mathrm{to}\:\mathrm{know}\:\mathrm{how}\:\mathrm{many}\:\mathrm{numbers} \\ $$$$\mathrm{that}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{have}\:\mathrm{2},\:\mathrm{3}\:\mathrm{or}\:\mathrm{5}\:\mathrm{as}\:\mathrm{factors}\: \\ $$$$\mathrm{1200}−\lfloor\frac{\mathrm{1200}}{\mathrm{2}}\rfloor=\mathrm{600}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1200} \\ $$$$\mathrm{600}−\lfloor\frac{\mathrm{600}}{\mathrm{3}}\rfloor=\mathrm{400}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{don}'\mathrm{t}\:\mathrm{have} \\ $$$$\mathrm{factor}\:\mathrm{3}\:\mathrm{or}\:\mathrm{2}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1200} \\ $$$$\mathrm{400}−\lfloor\frac{\mathrm{400}}{\mathrm{5}}\rfloor=\mathrm{320}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{doesn}'\mathrm{t}\: \\ $$$$\mathrm{have}\:\mathrm{factors}\:\mathrm{2},\:\mathrm{3}\:\mathrm{or}\:\mathrm{5}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1200} \\ $$$$\mathrm{Answer}:\:\mathrm{320} \\ $$

Commented by ZiYangLee last updated on 22/Oct/20

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