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Can some one explain how this is possible ?

    1)  13 / 9 = 1.(1 + 3) = 1.444 ...
    2)  23 / 9 = 2.(2 + 3) = 2.555 ...
    3)  35 / 9 = 3.(3 + 5) = 3.888 ...
    4)  47 / 9 = 4.(4 + 7) = 4.(11) → 4.(11 - 9) = 5.222 ...
    5)  63 / 9 = 6.(6 + 3) = 6.(9)  → 6.(9 - 9)  = 7.0 = 7
apnorton
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Sudheej
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1 Answers1

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Let $a$ be the tens digit and $b$ be the ones digit so that our two digit number is $10a+b$. In general, the pattern seems to be: $$ \dfrac{10a+b}{9} = (a+j).kkk... $$ where: $$(j,k)= \begin{cases} (0,a+b) & \text{if }a+b \in \{0,1,2,...,8\} \\ (1,a+b-9) & \text{if }a+b \in \{9,10,11,...,17\} \\ (2,a+b-18) & \text{if }a+b =18 \\ \end{cases}$$

In other words, $j$ and $k$ are the quotient and remainder (respectively) upon dividing $a+b$ by $9$. To see why this makes sense, recall that $\boxed{0.kkk... = k/9}$. This can be proven a number of ways (for example, it is a convergent geometric series). Hence, observe that: $$ \dfrac{10a+b}{9} =\dfrac{9a+(a+b)}{9}=a+\dfrac{a+b}{9}=a+j+\dfrac{k}{9} = (a+j).kkk... $$

Adriano
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