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

February 22, 2026

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.

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.

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

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.

Check: 98 ÷ 7 = 14 full cycles → each residue appears exactly 14 times from 1 to 98. But 99 = 7×14 + 1 → remainder 1 100 = 7×14 + 2 → remainder 2

So residue 3 occurs on k = 0 to 13 → 14 times — exactly one more than residue 4,5,6? Wait: k=0 →3, k=1→10, ..., k=13 →94. That’s 14 numbers.

Wait — but 7×14 + 3 = 101 > 100 → so max k is 13.

Hence, total = 14.

But wait: from k=0 to k=13 → that’s 14 terms.

Yes.

General Rule: How Many Integers ≤ N are ≡ a (mod m)?

The count is given by: ⌊(N − a) / m⌋ + 1 if a ≤ N

Apply with N = 100, a = 3, m = 7: (100 − 3) = 97 97 ÷ 7 = 13.857… → floor = 13 13 + 1 = 14

Alternatively, since each residue occurs ⌊N/m⌋ = 14 times, and remainder N mod m = 100 mod 7 = 2 → residues 1 and 2 occur extra time? No.

Actually, the distribution: In 1 to 98 (14×7), each residue 0–6 appears exactly 14 times.

Then:

99: 99 ÷ 7 = 14 × 7 = 98 → remainder 1 → so 99 ≡ 1 (mod 7) → residue 1 occurs 15 times
100: remainder 2 → residue 2 occurs 15 times

Wait — correction:

Let’s compute N = 100 m = 7 Quotient q = floor(99 / 7) = floor(99 ÷ 7) = 14 (since 7×14 = 98) Remainder r = 100 − 7×14 = 100 − 98 = 2

In the complete cycles 1–98, each residue 0–6 appears 14 times.

Now, for numbers 1 to 100:

Residues 1 and 2 get one extra value (99 and 100)
Residue 3? → 3, 10, ..., 94 → 14 values Next would be 101 → excluded So residue 3: 14 times

Wait — but what about residue 3? Number of times r ≡ a mod m occurs from 1 to N is: ⌊(N − a)/m⌋ + 1

So: ⌊(100 − 3)/7⌋ + 1 = ⌊97/7⌋ + 1 = 13 + 1 = 14

✓ Confirmed.

Conclusion

There are exactly 14 of the first 100 positive integers that are congruent to 3 modulo 7.

This elegant result stems from the predictable structure of integers modulo 7. Every block of 7 consecutive numbers contains exactly one number congruent to each residue 0–6. Among the first 98 numbers (14 full cycles), each residue appears 14 times. The 99th and 100th numbers add to residues 1 and 2 — so residue 3 remains at 14, the others gain at most one extra.

Understanding these patterns helps in cryptography, algorithm design, and number theory applications.

So next time you ask: How many of the first 100 positive integers are ≡ 3 (mod 7)? — the answer is clearly 14.

Keywords: congruent to 3 mod 7, first 100 integers, modular arithmetic, number theory, Counting residues, modular distribution, 3 mod 7, how many numbers ≡3 mod7, modular counting, mathematics education, integer patterns

Meta Description: Discover how many of the first 100 positive integers are congruent to 3 modulo 7. Learn the pattern, formula, and 14 precise answers using modular arithmetic.