The $\overline{abcdef}$ is a positive integer with distinct digits greater than 0 and lower than 7. If

- $\overline{ab}$ is divisible by 2,
- $\overline{abc}$ is divisible by 3,
- $\overline{abcd}$ is divisible by 4,
- $\overline{abcde}$ is divisible by 5,
- $\overline{abcdef}$ is divsible by 6.

How many possible values does $\overline{abcdef}$ have?

3
4
0
1
2

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Since $b, d, f$ are even numbers ( $2, 4, 6$ ) and $e$ is $5$ (because it can't be $0$ ), $a$ and $c$ are $1$ and $3$ in some order. $3\mid a+b+c$ , so $b$ has to be $6$ . If $d=4$ , then even $c=1$ or $c=3$ , $4\not\mid \overline{cd}$ , $\overline{abcd}$ , so $d=2$ . From that there are two possible solutions left, which are both correct:

Solution 1:$\overline{abcdef}=163254$Solution 2:$\overline{abcdef}=361254$