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

Answer 1

To solve $7?÷75=712?$, 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 $7x÷75=712y$, 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 $53400\mathrm{mod}7$:

$$53400÷7=7628\text{(with a remainder of 4)}$$

So, $53400\equiv 4\mathrm{mod}7$. To make $53400+75y$ a multiple of 7, we need:

$$4+75y\equiv 0\mathrm{mod}7$$

Next, we compute $75\mathrm{mod}7$:

$$75÷7=10\text{(with a remainder of 5)},\text{so}75\equiv 5\mathrm{mod}7$$

Thus, we need:

$$4+5y\equiv 0\mathrm{mod}7$$

which leads us to:

$$5y\equiv -4\equiv 3\mathrm{mod}7$$

Now we will test each $y$ from $0$ to $9$ to find values that satisfy $5y\equiv 3\mathrm{mod}7$:

• $y=0\to 5(0)\equiv 0$
• $y=1\to 5(1)\equiv 5$
• $y=2\to 5(2)\equiv 3$ (This works)
• $y=3\to 5(3)\equiv 1$
• $y=4\to 5(4)\equiv 6$
• $y=5\to 5(5)\equiv 4$
• $y=6\to 5(6)\equiv 2$
• $y=7\to 5(7)\equiv 0$
• $y=8\to 5(8)\equiv 5$
• $y=9\to 5(9)\equiv 3$ (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 $7$:

$$\text{Extra close brace or missing open brace}$$

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: $$\boxed{{\displaystyle 2}}$$