\[\large{\begin{eqnarray} 1 &=& 44 \div 44 \\ 2 &=& 4 \times 4 \div (4 + 4 ) \\ 3 &=& (4 + 4 + 4) \div 4 \\ 4 &=& 4 + (4\times(4-4)) \\ 5 &=& (4 + (4\times4)) \div 4 \end{eqnarray}} \]

Above shows the first 5 positive integers formed by using the four mathematical operators (\(+ \ - \ \times \ \div\)) only on the digit 4 four times.

What is the smallest positive integer that cannot be represented using these conditions?

Note: You are allowed to join the digits together: \(44 + 44 \).

\[\large 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8\ \Box \ 8 \ \Box\ 8=1000\]

What is the minimum number of operators that can be filled in the boxes to make the equation above true?

**Note:** Boxes can be left blank to denote concetanation of adjacent digits

\[1 \ \boxed{\phantom{0}} \ 2 \ \boxed{\phantom{0}} \ 3 \ \boxed{\phantom{0}} \ 4 \ \boxed{\phantom{0}} \ 5\]

Use the mathematical operators \(+,-,÷,×\) exactly once to fill in the blank boxes. What is the maximum real value that can be obtained?

Give your answer to two decimal places.

\[\large 1 \; \underbrace{\square \; 2 \; \square \; \cdots \; \square \; n \; \square}_{n \text{ number of }\square\text{'s}} \; (n+1) = n+ 2 \]

What is the minimum value of the positive integer \(n>1\) such that we can fill in all the boxes above by using at least one of the four mathematical operators ( \(+, -, \times , \div \) ) and the equation holds true?

**Note**: Order of operations (BODMAS) applied.

\[\large{\begin{eqnarray} 1 &=& 44 \div 44 \\ 2 &=& 4 \times 4 \div (4 + 4 ) \\ 3 &=& (4 + 4 + 4) \div 4 \\ 4 &=& 4 + (4\times(4-4)) \\ 5 &=& (4 + (4\times4)) \div 4 \end{eqnarray}} \]

Above shows the first 5 positive integers formed by using the four mathematical operators (\(+ \ - \ \times \ \div\)) only on the digit 4 four times.

What is the smallest positive integer that cannot be represented using these conditions?

Note: You are allowed to join the digits together: \(44 + 44 \).

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