10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\qquad$.

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【Analysis】Consider all lines drawn from the left end at intervals of 6 cm and 7 cm. Find the least common multiple (LCM) of 6 and 7, which is 42 cm, and see how many 3 cm segments are in each 42 cm.

【Solution】Solution: 240 is exactly divisible by 6, so "drawing a line every 6 cm from the right end" is equivalent to "drawing a line every 6 cm from the left end".
The LCM of 6 and 7 is: $6 \times 7 = 42$, so there is a cycle every 42 cm.
Analyze the cut points in one cycle: endpoint, 6 cm, 7 cm, 12 cm, 14 cm, 18 cm, 21 cm, 24 cm, 28 cm, 30 cm, 35 cm, 36 cm, 42 cm.
$21 - 18 = 3$ (cm), $24 - 21 = 3$ (cm),
So there are 2 segments of 3 cm in one cycle.
$240 \div 42 = 5$ (groups) $\cdots 30$ (cm),
There are $5 \times 2 = 10$ (segments) in 5 groups,
In the remaining 30 cm, there are 2 more segments of 3 cm, so there are $10 + 2 = 12$ segments of 3 cm in total;

Answer: There are 12 pieces of wood that are 3 cm long.
The answer is: 12.