What is the largest two-digit number that gives 4 as the remainder when divided by 7?

We are tasked with finding the largest two-digit number that gives a remainder of 4 when divided by 7.

Let's denote the number as \( N \). Mathematically, we express this as:
\[
N \equiv 4 \pmod{7}
\]
This means that \( N \) can be written as:
\[
N = 7k + 4
\]
where \( k \) is an integer.

### Step 1: Start from the largest two-digit number
The largest two-digit number is 99. We now find the largest \( N \leq 99 \) that satisfies \( N \equiv 4 \pmod{7} \).

### Step 2: Check numbers of the form \( N = 7k + 4 \)
We want \( N = 7k + 4 \leq 99 \), so:
\[
7k + 4 \leq 99
\]
\[
7k \leq 95
\]
\[
k \leq \frac{95}{7} \approx 13.57
\]
So, the largest integer \( k \) is 13.

### Step 3: Calculate the corresponding value of \( N \)
Now, substitute \( k = 13 \) into the equation \( N = 7k + 4 \):
\[
N = 7(13) + 4 = 91 + 4 = 95
\]

Thus, the largest two-digit number that gives a remainder of 4 when divided by 7 is \( \boxed{95} \).

Would you like more details on how remainders work, or have any questions about this?

Here are 5 related questions to explore:
1. What is the smallest two-digit number that gives 4 as the remainder when divided by 7?
2. How do you find the remainder when dividing larger numbers by a divisor?
3. Can you list all two-digit numbers that give a remainder of 4 when divided by 7?
4. What is the largest two-digit number divisible by 7?
5. How can you check if a number gives a specific remainder when divided by another number?

**Tip:** When solving remainder problems, always express the number in terms of the divisor to simplify finding patterns!

Discover the largest two-digit number that gives a remainder of 4 when divided by 7 with this clear step-by-step explanation. Learn how to apply modular arithmetic and algebra to solve remainder problems, suitable for students in Grades 6-8.

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