a problem of number theory
- HIHUMITHafiz
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Let N be a number with 9 distinct non-zero digits,such that ,for each k from 1 to 9 inclusive,the first k digits of N form a number which is divisible by k.Find N.
- Souvik saha
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Re: a problem of number theory
Let the number be $abcdefghi$
Clearly $e=5$.
Also $b,d,f,h$ are even. Hence $a,c,e,g,i$ odd.
$8|abcdefghi\Rightarrow 8|fgh\Rightarrow 8|gh$ Hence $h=2,6$ as $g$ is odd.
Also, $4|cd\Rightarrow d=2,6$ as $c$ is odd.
If $h=6$, $g=1,9$ Note that $abc$, $def$, $ghi$ must all be divisible by $3$. Hence $3|d+f+2, 3|g+i\Rightarrow g,i=3,9\Rightarrow g=9, i=3$.
$d=2\Rightarrow f=8$. So $abcdefghi=147258963,741258963$.
If $h=2$, $g=3,7$ then $3|d+f+2, 3|g+i+2$
$d=6\Rightarrow f=4\Rightarrow b=8$
a)$g=3$, $abcdefghi=789654321, 987654321, 189654327, 981654327$
b)$g=7$, $abcdefghi=189654723, 981654723, 183654729, 381654729$.
By checking, only $381654729$ satisfy.
Clearly $e=5$.
Also $b,d,f,h$ are even. Hence $a,c,e,g,i$ odd.
$8|abcdefghi\Rightarrow 8|fgh\Rightarrow 8|gh$ Hence $h=2,6$ as $g$ is odd.
Also, $4|cd\Rightarrow d=2,6$ as $c$ is odd.
If $h=6$, $g=1,9$ Note that $abc$, $def$, $ghi$ must all be divisible by $3$. Hence $3|d+f+2, 3|g+i\Rightarrow g,i=3,9\Rightarrow g=9, i=3$.
$d=2\Rightarrow f=8$. So $abcdefghi=147258963,741258963$.
If $h=2$, $g=3,7$ then $3|d+f+2, 3|g+i+2$
$d=6\Rightarrow f=4\Rightarrow b=8$
a)$g=3$, $abcdefghi=789654321, 987654321, 189654327, 981654327$
b)$g=7$, $abcdefghi=189654723, 981654723, 183654729, 381654729$.
By checking, only $381654729$ satisfy.
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- HIHUMITHafiz
- Posts:9
- Joined:Mon Aug 26, 2013 8:34 am
- Location:Mymensingh