a) На пять делятся числа вида 5n, n ∈ N.
Чисел от M до N, которые делятся на d будет:
([x] - целая часть)
Чисел от M до N, которые не делятся на d будет:
![(N - [\frac{N}d]) - (M - [\frac{M}d]) (N - [\frac{N}d]) - (M - [\frac{M}d])](https://tex.z-dn.net/?f=%28N+-+%5B%5Cfrac%7BN%7Dd%5D%29+-+%28M+-+%5B%5Cfrac%7BM%7Dd%5D%29+)
![m = [\frac{99}{5}] - [\frac{9}{5}] = 19 - 1 = 18\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{18}{90} = \frac{2}{10} = 0.2 m = [\frac{99}{5}] - [\frac{9}{5}] = 19 - 1 = 18\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{18}{90} = \frac{2}{10} = 0.2](https://tex.z-dn.net/?f=m+%3D+%5B%5Cfrac%7B99%7D%7B5%7D%5D+-+%5B%5Cfrac%7B9%7D%7B5%7D%5D+%3D+19+-+1+%3D+18%5C%5C%5C%5C+n+%3D+99+-+9+%3D+90%5C%5C%5C%5C+p+%3D+%5Cfrac%7Bm%7D%7Bn%7D+%3D+%5Cfrac%7B18%7D%7B90%7D+%3D+%5Cfrac%7B2%7D%7B10%7D+%3D+0.2)
![b) \ m = [\frac{99}{13}] - [\frac{9}{13}] = 7 - 0 = 7\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{7}{90} b) \ m = [\frac{99}{13}] - [\frac{9}{13}] = 7 - 0 = 7\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{7}{90}](https://tex.z-dn.net/?f=b%29+%5C+m+%3D+%5B%5Cfrac%7B99%7D%7B13%7D%5D+-+%5B%5Cfrac%7B9%7D%7B13%7D%5D+%3D+7+-+0+%3D+7%5C%5C%5C%5C+n+%3D+99+-+9+%3D+90%5C%5C%5C%5C+p+%3D+%5Cfrac%7Bm%7D%7Bn%7D+%3D+%5Cfrac%7B7%7D%7B90%7D)
c) Чисел от M до N, которые делятся или на d1 или на d2 будет:![( [\frac{N}{d1}] - [\frac{M}{d1}]) + ( [\frac{N}{d2}] - [\frac{M}{d2}]) - ( [\frac{N(d1,d2)}{d1d2}] - [\frac{M(d1,d2)}{d1d2}]) ( [\frac{N}{d1}] - [\frac{M}{d1}]) + ( [\frac{N}{d2}] - [\frac{M}{d2}]) - ( [\frac{N(d1,d2)}{d1d2}] - [\frac{M(d1,d2)}{d1d2}])](https://tex.z-dn.net/?f=%28+%5B%5Cfrac%7BN%7D%7Bd1%7D%5D+-+%5B%5Cfrac%7BM%7D%7Bd1%7D%5D%29+%2B+%28+%5B%5Cfrac%7BN%7D%7Bd2%7D%5D+-+%5B%5Cfrac%7BM%7D%7Bd2%7D%5D%29+-+%28+%5B%5Cfrac%7BN%28d1%2Cd2%29%7D%7Bd1d2%7D%5D+-+%5B%5Cfrac%7BM%28d1%2Cd2%29%7D%7Bd1d2%7D%5D%29)
((d1,d2) - НОД)
![m = ( [\frac{99}{15}] - [\frac{9}{15}]) + ( [\frac{99}{25}] - [\frac{9}{25}]) - ( [\frac{99*5}{15*25}] - [\frac{9*5}{15*25}]) =\\\\ 6 + 3 - 1 = 8\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{8}{90} = \frac{4}{45} m = ( [\frac{99}{15}] - [\frac{9}{15}]) + ( [\frac{99}{25}] - [\frac{9}{25}]) - ( [\frac{99*5}{15*25}] - [\frac{9*5}{15*25}]) =\\\\ 6 + 3 - 1 = 8\\\\ n = 99 - 9 = 90\\\\ p = \frac{m}{n} = \frac{8}{90} = \frac{4}{45}](https://tex.z-dn.net/?f=m+%3D+%28+%5B%5Cfrac%7B99%7D%7B15%7D%5D+-+%5B%5Cfrac%7B9%7D%7B15%7D%5D%29+%2B+%28+%5B%5Cfrac%7B99%7D%7B25%7D%5D+-+%5B%5Cfrac%7B9%7D%7B25%7D%5D%29+-+%28+%5B%5Cfrac%7B99%2A5%7D%7B15%2A25%7D%5D+-+%5B%5Cfrac%7B9%2A5%7D%7B15%2A25%7D%5D%29+%3D%5C%5C%5C%5C+6+%2B+3+-+1+%3D+8%5C%5C%5C%5C+n+%3D+99+-+9+%3D+90%5C%5C%5C%5C+p+%3D+%5Cfrac%7Bm%7D%7Bn%7D+%3D+%5Cfrac%7B8%7D%7B90%7D+%3D+%5Cfrac%7B4%7D%7B45%7D+)
d) p(не делится на двадцать девять) = 1 - p(делится на двадцать девять)
![m = ( [\frac{99}{29}] - [\frac{9}{29}]) + ( [\frac{99}{29}] - [\frac{9}{29}]) = 3\\\\ n = 99 - 9 = 90\\\\ p = 1 - \frac{m}{n} = 1 - \frac{3}{90} = \frac{29}{30} m = ( [\frac{99}{29}] - [\frac{9}{29}]) + ( [\frac{99}{29}] - [\frac{9}{29}]) = 3\\\\ n = 99 - 9 = 90\\\\ p = 1 - \frac{m}{n} = 1 - \frac{3}{90} = \frac{29}{30}](https://tex.z-dn.net/?f=m+%3D+%28+%5B%5Cfrac%7B99%7D%7B29%7D%5D+-+%5B%5Cfrac%7B9%7D%7B29%7D%5D%29+%2B+%28+%5B%5Cfrac%7B99%7D%7B29%7D%5D+-+%5B%5Cfrac%7B9%7D%7B29%7D%5D%29+%3D+3%5C%5C%5C%5C+n+%3D+99+-+9+%3D+90%5C%5C%5C%5C+p+%3D+1+-+%5Cfrac%7Bm%7D%7Bn%7D+%3D+1+-+%5Cfrac%7B3%7D%7B90%7D+%3D+%5Cfrac%7B29%7D%7B30%7D+)