How many of the first 100 positive integers are congruent to 3 (mod 7)? - Aurero
Title: 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.
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
🔗 Related Articles You Might Like:
📰 Spawn Movie Is SHOCKING! Cult Classic Meets Impact in This Studio Blockbuster! 📰 Why the Spawn Movie Breaked Box Office and Fans Alike—Spoiler Alert! 📰 Spawn Movie Explosion: The Movie That Cut Deep and Stood the Test of Time 📰 Veuve Unveiled The Hidden Inspiration Youve Never Heard Before Inside Her Iconic Legacy 📰 Veuves Devotees Are In Astonishmentwhat This Hidden Truth Reveals About Her Mystery Allure 📰 Veuves Untold Story The Shocking Choice That Redefined Her Timeless Concept Forever 📰 Vibe Dispensary Attack Why This Bulletproof Space Changed My Whole Mood 📰 Vickie Lynn Hogan Finally Spills The Truth No One Expected 📰 Vickie Lynn Hogan Reveals The Hidden Secret No One Talked About 📰 Vicksburg Ms Secrets Hidden Beneath The Soil You Never Knew Existed 📰 Victoria Beer Is Hiding Secrets Youve Never Seendrink Up And See What Happens 📰 Victoria Beer The Only Drink You Didnt Know You Needed 📰 Victoria Justice Exposed Nude Images Rock Hollywood Crash Her Career 📰 Victoria Justice Stunned Scandalous Picture Shatters Her Innocence 📰 Victoria Justices Secret Leaked What She Said Public Shock Stuns Fans 📰 Victoria Mboko Exposed The Alarming Revelation About Her Past That Barks Silence 📰 Victoria Mbokos Shocking Secret Behind Her Rise To Fame No One Saw Coming 📰 Victoria Mbokos Silence Speaks Volumes The Shocking Story No One Told YouFinal 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.
