a such that 10^n+a|10^{n+1}+a

kunny

For positive integer $n$ , find all positive integers $a$ such that $0<a<10^n$ and $10^n+a|10^{n+1}+a$ in terms of $n$ .

**Posted:** Sat Nov 07, 2009 5:55 am

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Today's calculation of Integral Digest

grafitti123 · P versus NP Offline Joined: 24 Apr 2009 Posts: 47

Consider $n > 2$
$10^{n + 1} + a = k(10^{n} + a)$
$10^{n}\frac {(10 - k)}{(k - 1)} = a$
$k$ can take values from 2 to 9.
Since $a < 10^{n}$ it implies $10 - k < k - 1$
This implies $k$ can take values from $6$ to $9$ . Since $n > 2$ the only values which satisfy $k$ are $6,7,9$
So $a$ can take values $(2^{n + 2} \cdot 5^{n - 1}) (2^{n - 1} \cdot 5^{n}) (2^{n - 3} \cdot 5^{n})$

**Posted:** Sat Nov 07, 2009 7:06 am