# Laba jeer da'da laba qof1017

Tixgeli laba qof $$A$$ iyo $$B$$ oo aan dhalan isla maalin iyo $$A$$ kayar $$B$$ . Muuji: Dhab ahaan waxa jira laba xiddigood oo da 'ah $$a,b \in \mathbb{N}$$ , oo khuseeya: $$2\cdot a = b$$ Waxaan marka hore dejinay $$d \in \mathbb{R}^+$$ sida farqiga da'da ee u dhexeeya $$A$$ iyo $$B$$ dhalashada $$A$$ leh $$d = d_0 + d_1$$ , $$d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[$$ . Waxaan hadda tixgelinaynaa qodob aan macquul ahayn waqtiga $$x \in \mathbb{R}^+$$ dhalashada $$A$$ kadib $$x = x_0 + x_1$$ , $$x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[$$ .

Waqtigan xaadirka ah, qeexitaan ahaan, $$a = \lfloor x \rfloor$$ iyo $$b = \lfloor x+d \rfloor$$ . Waxaan hadda go'aamineynaa dhammaan $$x$$ wixii xajinaya:

$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$

Kiiska 1aad: $$0 \leq x_1 + d_1 < 1$$:

Kadib $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$

Tani waxay ka dhigan tahay in $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ iyo $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ da'da koowaad $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .

Kiiska 2aad: $$1 \leq x_1 + d_1 < 2$$:

Kadib $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$

Tani waxay ka dhigan tahay in $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ iyo $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ kooxda labaad ee da 'da.

Gaar ahaan, tan macnaheedu waa, tusaale ahaan: Haddii hooyadaa kugu dhasho da'da $$20$$ sano, waxay dhab ahaan labanlaab tahay da'daada $$40$$ iyo $$42$$ sano. Sidoo kale waa wax xiiso leh haddii iyo goorta ay $$n$$ jeer ay duug tahay: Halkan $$n \in \mathbb{N}$$ si aan macquul ahayn oo waxaan helnaa $$x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)$$ . Tani waxay si sax ah u shaqeysaa marka farqiga da'da isku dhafan $$\lfloor d \rfloor = d_0$$ tiro badan tahay $$n-1$$ , tusaale ahaan kiiska kore hooyadaadu waa $$24$$ sano $$6$$ jeer da'daada.

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