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Olympiad workbook
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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let α and β be positive integers such that (16)/(37) < (α)/(β) <...

Let α\alpha and β\beta be positive integers such that 1637<αβ<716\frac{16}{37} < \frac{\alpha}{\beta} < \frac{7}{16}. Find the smallest possible value of β\beta.

Correct answer

23

Step-by-step solution

∵1637<αβ<716⇒3716>βα>167⇒3716α>β>167α\begin{aligned} & \because \quad \frac{16}{37} < \frac{\alpha}{\beta} < \frac{7}{16} \\ & \Rightarrow \quad \frac{37}{16}>\frac{\beta}{\alpha}>\frac{16}{7} \\ & \Rightarrow \quad \frac{37}{16} \alpha>\beta>\frac{16}{7} \alpha\end{aligned} ⇒27α<β−2α<516α,β−2α∈I and lie between (2α7,516α)≡(32α112,35α112)⇒(β−2α)∈(0.28α,0.31α) If α=10⇒(2.8,3.1)⇒β−2α=3⇒β=23\begin{aligned} & \Rightarrow \quad \frac{2}{7} \alpha < \beta-2 \alpha < \frac{5}{16} \alpha, \beta-2 \alpha \in I \text { and lie between }\left(\frac{2 \alpha}{7}, \frac{5}{16} \alpha\right) \equiv\left(\frac{32 \alpha}{112}, \frac{35 \alpha}{112}\right) \\ & \Rightarrow \quad(\beta-2 \alpha) \in(0.28 \alpha, 0.31 \alpha) \\ & \text { If } \alpha=10 \Rightarrow(2.8,3.1) \\ & \Rightarrow \quad \beta-2 \alpha=3 \\ & \Rightarrow \quad \beta=23\end{aligned}
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