10. (6 points) The number of three-digit numbers that can be divided by 5 and 6, and contain at least one 6 is $\qquad$.

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Note: The note at the end is not part of the translation but is provided for clarity. The actual translation is above.
【Solution】Solution: According to the analysis, decompose the prime factors $6=2 \times 3$
$\therefore$ This three-digit number can be simultaneously divisible by $2, 3, 5$, and the number must contain at least one 6.
$\therefore$ The unit digit of this three-digit number must be even or 0, and since the unit digit of a number divisible by 5 must be 0 or 5, the unit digit is 0,

Let this three-digit number be $\overline{\mathrm{ab} 0}$, according to the problem, at least one of $a, b$ must be 6,
(1) When $a=6$, then $6+b+0$ is a multiple of 3, so $b=0,3,6,9$, the valid three-digit numbers are: $600, 630, 660, 690$
(2) When $b=6$, then $6+a+0$ is a multiple of 3, so $a=3,6,9$, the valid three-digit numbers are: $360, 660, 960$ In summary, the three-digit numbers that meet the conditions are: $360, 660, 960, 600, 630, 690$ Therefore, the answer is: 6.