a-and-b-are-mutuallly-prime-numbers-If-3-8-lt-a-b-lt-2-5-Find-b-a-min-

Question Number 181489 by Shrinava last updated on 25/Nov/22

a and b are mutuallly prime numbers

If \frac{3}{8} < \frac{a}{b} < \frac{2}{5}

Find {(b - a)}_{\min}

Answered by mr W last updated on 26/Nov/22

\frac{3}{8} < \frac{a}{b} < \frac{2}{5}

\frac{3 b}{8} < a < \frac{2 b}{5}

⌊ \frac{3 b}{8} ⌋ + 1 ⩽ a ⩽ ⌈ \frac{2 b}{5} ⌉ - 1

⌊ \frac{3 b}{8} ⌋ + 1 ⩽ ⌈ \frac{2 b}{5} ⌉ - 1

⌈ \frac{2 b}{5} ⌉ - ⌊ \frac{3 b}{8} ⌋ ⩾ 2

\Rightarrow \frac{a}{b} = \frac{5}{13}, \frac{7}{18}, \frac{8}{21}, \frac{9}{23}, \frac{11}{28}, \frac{11}{29}, \frac{12}{31}, \dots

{(b - a)}_{m i n} = 13 - 5 = 8

Commented by Shrinava last updated on 28/Nov/22

cool dear professor thank you