The largest three-digit number that gives the same remainder 2 when divided by 3, 5, and 9 is ______.

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**Solution:**

1. Find the LCM of 3, 5, and 9:
   - Prime factorization:
     - 3 = $3^1$
     - 5 = $5^1$
     - 9 = $3^2$
   - LCM = $3^2 	imes 5 = 9 	imes 5 = 45$

2. The number must leave the same remainder 2 when divided by 3, 5, and 9.
   - This means the number is of the form:
     $$	ext{Number} = 45k + 2$$

3. Find the largest three-digit number of this form:
   - Largest three-digit number is 999.
   - Find the largest multiple of 45 less than or equal to 999:
     $$999 \div 45 = 22 	ext{ remainder } 9$$
     So, largest multiple of 45 is:
     $$45 	imes 22 = 990$$

4. Add 2 to get the required number:
   $$990 + 2 = 992$$

5. Verify:
   - $992 \div 3$ leaves remainder 2
   - $992 \div 5$ leaves remainder 2
   - $992 \div 9$ leaves remainder 2

**Answer:** $992$

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