Interesting Multiples

Number Theory Level 4

For how many positive integers n where \( 1 \le n \le 1000 \) does there exist some multiple of n which is a sequence of 1s followed by a sequence of 0s in base 10?

\( \textbf{ Details and Examples } \)

There must be at least one 1 at the beginning of the multiple and it must end in at least one 0

10 is such an n since it itself satisfies the conditions

6 is such an n since \( 6 \times 185 = 1110 \) which also satisfies the conditions

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