Understanding and Solving: Counting How Many of the First 100 Positive Integers Satisfy a Given Congruence

When it comes to number theory in mathematics, congruences play a vital role—especially in problems involving modular arithmetic. A common challenge often presented is:
How many of the first 100 positive integers satisfy a particular congruence condition?

While the exact congruence isn’t specified, this article explores a general solution approach using modular arithmetic, walk through practical examples, and provides methods to efficiently count solutions within a finite range—such as the first 100 positive integers.

Understanding the Context

What Is a Congruence?

A congruence expresses whether two integers leave the same remainder when divided by a positive integer (the modulus). For example:
x ≡ a (mod n) means that x and a leave the same remainder upon division by n, or equivalently, n divides (x − a).

In this context, we are interested in counting integers x in the set {1, 2, 3, ..., 100} such that:
x ≡ a (mod n) for fixed integers a and n.

Solution: We are asked to count how many of the first 100 positive integers satisfy the congruence:

Key Insights

Example Problem

Let’s suppose the problem asks:
How many of the first 100 positive integers are congruent to 3 modulo 7?
That is, find the count of integers x such that:
x ≡ 3 (mod 7), and 1 ≤ x ≤ 100

Step-by-Step Solution

Continue Reading

The sum of an arithmetic series is 200. The first term is 5, and the last term is 45. How many terms are there?
Sum of arithmetic series \(S = \frac{n}{2}(a + l)\), where \(a = 5\), \(l = 45\), \(S = 200\).
\(200 = \frac{n}{2}(5 + 45) = \frac{n}{2}(50)\).

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Final Thoughts

  1. Understand the Pattern of Solutions
    The general solution to x ≡ 3 (mod 7) is:
    x = 7k + 3, where k is any integer

  2. Find Valid Values of k
    We need 1 ≤ 7k + 3 ≤ 100

Solve for k:
1 ≤ 7k + 3 ⇒ 7k ≥ –2 ⇒ k ≥ 0 (since k must be integer)
7k + 3 ≤ 100 ⇒ 7k ≤ 97 ⇒ k ≤ ⌊97/7⌋ = 13

So k ranges from 0 to 13 inclusive.

  1. Count the Valid k Values
    k = 0, 1, 2, ..., 13 → total of 14 values

Thus, there are 14 integers between 1 and 100 that satisfy x ≡ 3 (mod 7).

General Strategy for Counting Solutions (1 ≤ x ≤ 100)

For a congruence x ≡ a (mod n), follow these steps:

  1. Express solution set:
    x = n·k + a, where k is an integer