How many of the first 100 positive integers are congruent to 3 (mod 7)?

Title: How Many of the First 100 Positive Integers Are Congruent to 3 (mod 7)?

Understanding number modular arithmetic can reveal fascinating patterns—one interesting question is: How many of the first 100 positive integers are congruent to 3 modulo 7? This seemingly simple inquiry uncovers how evenly integers spread across residue classes and highlights the predictable patterns in modular systems.

Understanding the Context

What Does “Congruent to 3 mod 7” Mean?

When we say an integer n is congruent to 3 modulo 7, we write:
n ≡ 3 (mod 7)

This means n leaves a remainder of 3 when divided by 7. In other words, n can be expressed in the form:
n = 7k + 3
where k is a non-negative integer.

How many of the first 100 positive integers are congruent to 3 (mod 7)?

Key Insights

Finding All Numbers ≤ 100 Such That n ≡ 3 (mod 7)

We want to count how many values of n in the range 1 to 100 satisfy n = 7k + 3.

Start by solving:
7k + 3 ≤ 100
7k ≤ 97
k ≤ 13.857…

Since k must be an integer, the largest possible value is k = 13.
Now generate the sequence:

For k = 0 → n = 7(0) + 3 = 3
k = 1 → n = 10
k = 2 → n = 17
...
k = 13 → n = 7(13) + 3 = 94 + 3 = 94 + 3? Wait: 7×13 = 91 → 91 + 3 = 94

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Final Thoughts

Wait: 7×13 = 91 → n = 91 + 3 = 94
k = 14 → 7×14 + 3 = 98 + 3 = 101 > 100 → too big

So valid values of k go from 0 to 13 inclusive → total of 14 values.

List the Numbers (Optional Verification):

The numbers are:
3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94

Count them — indeed 14 numbers.

Why Does This Pattern Occur?

In modular arithmetic with modulus 7, the possible residues are 0 through 6. When dividing 100 numbers, each residue class mod 7 appears approximately 100 ÷ 7 ≈ 14.28 times.

Specifically, residues 0 to 3 mod 7 occur 15 times in 1–98 (since 98 = 14×7), and then residues 4–6 only appear 14 times by 98. However, residue 3 continues into 100:
Indeed, n = 3, 10, ..., 94, and the next would be 101 — outside the range.