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Step-by-Step Solution
Step 1: Understand the Sets Involved
We are given two sets:
• S = \{4,\,6,\,9\}
• T = \{9,\,10,\,11,\,\ldots,\,1000\}
We define set A as all possible sums of one or more elements from S , i.e.
A = \{a_{1} + a_{2} + \ldots + a_{k} \ | \ k \in \mathbf{N}, \ a_{i} \in S\ \text{for all}\ i\} .
Step 2: Expressing Numbers as Linear Combinations of 4, 6, and 9
Each element in A can be written in the form 4x + 6y + 9z with x, y, z \ge 0 (where x, y, z are non-negative integers).
Step 3: Identify the Range of Expressible Numbers
We need to check from which point onwards every number can be expressed as 4x + 6y + 9z . A useful idea here is to look at smaller combinations first:
• 6y + 9z = 3(2y + 3z) . Since 2 and 3 are coprime, all integers from 2 onwards can be expressed in the form 2y + 3z . Hence, all multiples of 3 (starting from 3 \cdot 2 = 6 ) can be formed by 6y + 9z .
• Next, consider adding 4x to these multiples of 3. Since 4 and 3 are coprime, once you can express every integer above a certain point as a combination of 4 and 3 , all large enough numbers can be written as 4x + 3k . The largest number that cannot be formed by 4 and 3 is 4 \times 3 - 4 - 3 = 5 , so from 6 onwards, every integer can be written as 4x + 3(\cdot) .
Combining these facts, it follows that every integer from 12 onwards can be expressed as 4x + 6y + 9z .
Step 4: Check Smaller Numbers in T = {9, 10, 11, …, 1000}
We need to see if any numbers less than 12 (but at least 9, since T starts from 9) fail to be written in the form 4x + 6y + 9z .
• 9 can be formed as 9 \times 1 + 0 + 0 , i.e. 9(1) .
• 10 can be formed as 4(1) + 6(1) .
• 11 cannot be expressed as 4x + 6y + 9z with non-negative integers x, y, z . You can verify by trying x = 0,1,2,\dots but it will not yield 11 .
• 12 can be formed, for example, as 4(3) or 9 + 3 (when x=0, y=1, z=1 for 6y + 9z plus an extra 0 from 4, or as 4 \times 3 ).
Step 5: Determine the Set A and T - A
Since every number from 9 upward (except 11 ) can be expressed as a sum of elements in S , it follows that:
• A contains all integers from 9 to 1000 , except 11 .
• Therefore, T - A = \{11\} .
Step 6: Sum of Elements in T - A
The set T - A only contains the element 11 . Hence, the sum of all elements in T - A is:
11 .
Final Answer
The required sum is 11 .
