 How many positive integers $N$ from $1$ to $5000$ satisfy the congruence $N \equiv 4 \pmod{7}$? 

We are tasked with finding how many positive integers $ N $ from 1 to 5000 satisfy the congruence:

\[
N \equiv 4 \pmod{7}
\]

### Step 1: General form of numbers satisfying the congruence

The integers $ N $ that satisfy $ N \equiv 4 \pmod{7} $ can be expressed in the form:
\[
N = 7k + 4
\]
where $ k $ is a non-negative integer. This is because for any integer $ k $, $ N $ will leave a remainder of 4 when divided by 7.

### Step 2: Find the range of $ N $ from 1 to 5000

We need to find the values of $ k $ such that:
\[
1 \leq N = 7k + 4 \leq 5000
\]
First, solve for $ k $:
\[
1 \leq 7k + 4 \leq 5000
\]
Subtract 4 from all parts of the inequality:
\[
-3 \leq 7k \leq 4996
\]
Now, divide the entire inequality by 7:
\[
-\frac{3}{7} \leq k \leq \frac{4996}{7}
\]
This simplifies to:
\[
0 \leq k \leq 714
\]
Thus, $ k $ can range from 0 to 714, inclusive.

### Step 3: Count the number of possible values of $ k $

The possible values of $ k $ are the integers from 0 to 714. The total number of such values is:
\[
714 - 0 + 1 = 715
\]

Thus, there are $ \boxed{715} $ positive integers $ N $ from 1 to 5000 that satisfy $ N \equiv 4 \pmod{7} $.

Would you like to explore more details or ask any follow-up questions? Here are five related questions you might consider:

1. How would the solution change if the modulus were different (e.g., $ N \equiv 4 \pmod{9} $)?
2. Can we generalize this approach for any modulus $ m $?
3. How can we solve similar problems with inequalities in different ranges?
4. What is the significance of modular arithmetic in number theory?
5. How does modular arithmetic relate to solving Diophantine equations?

**Tip:** When working with modular arithmetic, expressing numbers in the form $ N = mk + r $ (where $ r $ is the remainder) simplifies finding solutions in a range.

How many positive integers $N$ from $1$ to $5000$ satisfy the congruence $N \equiv 4 \pmod{7}$?

Explore how to find the number of integers N from 1 to 5000 that satisfy N ≡ 4 (mod 7). This problem uses key number theory concepts and modular arithmetic, providing a step-by-step solution suitable for Grades 9-12.

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