For each positive integer n, find the number of positive integers that are less than 210n which are odd multiples of three that are not multiples of five and are not multiples of seven. Justify your answer, which should be in terms of n.

Question
Discrete math
asked 2021-01-19
For each positive integer n, find the number of positive integers that are less than 210n which are odd multiples of three that are not multiples of five and are not multiples of seven. Justify your answer, which should be in terms of n.

Answers (1)

2021-01-20
Total multiply of k less than \(\displaystyle{n}={\left[{\frac{{{n}-{1}}}{{{k}}}}\right\rbrace}\)
a) Multiply of \(\displaystyle{3}={\left[{\frac{{{210}{n}-{1}}}{{{3}}}}\right]}={\left[{\frac{{{70}{n}-{1}}}{{{3}}}}\right]}\)
\(\displaystyle={70}{n}\)
Odd multiply of \(\displaystyle{3}={\frac{{{70}{n}}}{{{2}}}}={35}{n}\)
b) Odd multiply of 3 and 5
- Total - even
\(\displaystyle={\left[{\frac{{{210}{n}-{1}}}{{{3},{9}}}}\right]}-{\left[{\frac{{{210}{n}-{1}}}{{{2},{3},{9}}}}\right]}={14}{n}-{7}{n}\)
\(\displaystyle={7}{n}\)
c) Odd multiply of 3 and 7
\(\displaystyle={\left[{\frac{{{210}{n}-{1}}}{{{3},{7}}}}\right]}-{\left[{\frac{{{210}{n}-{1}}}{{{2},{3},{7}}}}\right]}={10}{n}-{5}{n}\)
\(\displaystyle{5}{n}\)
d) Odd multiply of 3,5 and 7
\(\displaystyle{\left[{\frac{{{210}{n}-{1}}}{{{3},{7}}}}\right]}-{\left[{\frac{{{210}{n}-{1}}}{{{2},{3},{5},{7}}}}\right]}={2}{n}-{n}={n}\)
Answer: \(\displaystyle-{35}{n}-{7}{n}-{5}{n}+{n}={24}{n}\)
0

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