We are given that the number $86A4$ is divisible by 7, and we need to find how many values of $A$ make the number divisible by 7.

Step 1: Express the number

The number can be written as: $86A4 = 8600 + 10A + 4$ This is the number $8600 + 10A + 4 = 8604 + 10A$.

Step 2: Check divisibility by 7

For divisibility by 7, we need to check when $8604 + 10A$ is divisible by 7. In other words, we need to solve: $8604 + 10A \equiv 0 \pmod{7}$

Step 3: Find the remainder when 8604 is divided by 7

Let's first find $8604 \mod 7$.

$8604 \div 7 = 1228 \text{ with a remainder of } 8604 - (1228 \times 7) = 8604 - 8600 = 4$ So, $8604 \equiv 4 \pmod{7}$.

Step 4: Solve for $A$

We need to solve the congruence: $8604 + 10A \equiv 0 \pmod{7}$ Substitute $8604 \equiv 4 \pmod{7}$: $4 + 10A \equiv 0 \pmod{7}$ Simplify $10A \pmod{7}$: $10A \equiv 3A \pmod{7}$ So the equation becomes: $4 + 3A \equiv 0 \pmod{7}$ Now subtract 4 from both sides: $3A \equiv -4 \pmod{7}$ Since $-4 \equiv 3 \pmod{7}$, we have: $3A \equiv 3 \pmod{7}$

Step 5: Solve for $A$

To isolate $A$, divide both sides by 3. Since 3 and 7 are coprime, division by 3 is valid, and we get: $A \equiv 1 \pmod{7}$ Thus, $A = 1$ is the only value that satisfies the congruence.

Conclusion:

The only value of $A$ that makes $86A4$ divisible by 7 is $A = 1$.

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Would you like a detailed breakdown of the modular arithmetic, or do you have any further questions about divisibility rules?

Here are some related questions for further exploration:

1. How do you determine divisibility by numbers other than 7?
2. What is the process for solving modular equations like $3A \equiv 3 \pmod{7}$?
3. How can we check divisibility by 11 for a number like $86A4$?
4. What happens if $A$ is replaced by a different variable, say $B$, in the number $86B4$?
5. Can you explain the process of checking divisibility for large numbers, say for numbers with 6 digits?

Tip: When dealing with modular arithmetic, always reduce terms before simplifying the equation.