Find the next date when all three community training programs, repeating every 10, 15, and 20 days respectively, will occur on the same day.

Here are the solutions for Item 6 and Item 7:

Item 6: Community Training Programs

a) Find the next date when all three programs will occur on the same day.

Step 1: Find the Least Common Multiple (LCM) of the days each program repeats.
The programs repeat after 10, 15, and 20 days.
Prime factorization:

$$10 = 2 \times 5$$ $$15 = 3 \times 5$$ $$20 = 2^2 \times 5$$

The LCM is found by taking the highest power of each prime factor:

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