Question

What is the missing number in this simplified exponential expression: 7?÷75=712 ?

Answer 1

To solve \( 7? \div 75 = 712 ? \), we first need to determine what \(7?\) and \(712?\) represent.

Let's denote \(7?\) as \(7x\) where \(x\) is the missing digit in the first part, and \(712?\) as \(712y\) where \(y\) is the missing digit in the second part.

From the equation \(7x \div 75 = 712y\), we can rewrite it as:

\[
7x = 75 \times 712y
\]

Now let's calculate \(75 \times 712\):

\[
75 \times 712 = 53400
\]

Thus, the equation simplifies to:

\[
7x = 53400 + 75y
\]

Next, we can express \(x\) and \(y\) as single digits (0-9). This means that \(53400 + 75y\) must be a multiple of 7.

Now, we can calculate \(53400 \mod 7\):

\[
53400 \div 7 = 7628 \quad \text{(with a remainder of 4)}
\]

So, \(53400 \equiv 4 \mod 7\). To make \(53400 + 75y\) a multiple of 7, we need:

\[
4 + 75y \equiv 0 \mod 7
\]

Next, we compute \(75 \mod 7\):

\[
75 \div 7 = 10 \quad \text{(with a remainder of 5)}, \quad \text{so} \quad 75 \equiv 5 \mod 7
\]

Thus, we need:

\[
4 + 5y \equiv 0 \mod 7
\]

which leads us to:

\[
5y \equiv -4 \equiv 3 \mod 7
\]

Now we will test each \(y\) from \(0\) to \(9\) to find values that satisfy \(5y \equiv 3 \mod 7\):

- \(y = 0 \rightarrow 5(0) \equiv 0\)
- \(y = 1 \rightarrow 5(1) \equiv 5\)
- \(y = 2 \rightarrow 5(2) \equiv 3\) (This works)
- \(y = 3 \rightarrow 5(3) \equiv 1\)
- \(y = 4 \rightarrow 5(4) \equiv 6\)
- \(y = 5 \rightarrow 5(5) \equiv 4\)
- \(y = 6 \rightarrow 5(6) \equiv 2\)
- \(y = 7 \rightarrow 5(7) \equiv 0\)
- \(y = 8 \rightarrow 5(8) \equiv 5\)
- \(y = 9 \rightarrow 5(9) \equiv 3\) (This works again)

Thus, potential values for \(y\) are \(2\) and \(9\).

Next, let's check \(y = 2\) (as \(2\) is in the smaller range). So, we have \(7122\).

To find \(x\), substitute \(y = 2\) back into \(7x\):

\[
7x = 53400 + 75(2)
\]

Calculating \(75 \times 2\) gives \(150\).

Now we find:

\[
7x = 53400 + 150 = 53550
\]

Dividing both sides by \(7\):

\[
x = \frac{53550}{7} = 7643 \quad \text{(implying that } 7643 \text{ should be attainable by } x)}
\]

This means for \(y = 2\), we could possibly derive \(x = 7\). Thus indicating:

In conclusion, the result for the expression is:

\(x = 7\) and \(y = 2\), and the missing numbers are \(7\) and \(2\). Thus \(? = 2\).

So the missing digit in the context is:
\[
\boxed{2}
\]