Find the smallest three-digit number congruent to 4 or 7 modulo 9.
["Find the Smallest Three-Digit Number Congruent to 4 or 7 Modulo 9 – A Step-by-Step Guide", "When exploring modular arithmetic, one common question is: What is the smallest three-digit number congruent to 4 or 7 modulo 9? This query is key for students, mathematicians, or anyone interested in number theory and cyclic patterns in integers. In this article, we’ll break down how to find the smallest three-digit number that satisfies either of these congruences efficiently and clearly.", "---", "### Understanding Modulo 9 and Congruences", "Before diving into the solution, let’s clarify what it means to be congruent modulo 9. Two numbers are congruent modulo 9 if they leave the same remainder when divided by 9.", "For example:\n- ( x \equiv 4 \pmod{9} ) means ( x = 9k + 4 ) for some integer ( k )\n- ( x \equiv 7 \pmod{9} ) means ( x = 9k + 7 )", "We are seeking the smallest three-digit number (i.e., ≥ 100) such that ( x \equiv 4 \pmod{9} ) or ( x \equiv 7 \pmod{9} ).", "---", "### Step 1: Find the smallest three-digit number ≥ 100", "The smallest three-digit number is 100.", "We want the smallest ( x \geq 100 ) where:\n- ( x \equiv 4 \pmod{9} )\n- or ( x \equiv 7 \pmod{9} )", "---", "### Step 2: Test if 100 satisfies either congruence", "Divide 100 by 9:\n( 100 \div 9 = 11 ) remainder ( 1 ) → So,\n( 100 \equiv 1 \pmod{9} )", "Since 1 is neither 4 nor 7, 100 is not our number.", "---", "### Step 3: Find the next number congruent to 4 or 7 mod 9", "We now look for the smallest number ≥ 100 such that ( x \equiv 4 \pmod{9} ) or ( x \equiv 7 \pmod{9} ).", "We can calculate the smallest ( x \geq 100 ) satisfying each congruence separately.", "#### For ( x \equiv 4 \pmod{9} ):", "Find the smallest ( k ) such that:\n( 9k + 4 \geq 100 )", "Solve:\n( 9k \geq 96 ) → ( k \geq 96/9 = 10.\overline{6} )\nSo smallest integer ( k = 11 )", "Then:\n( x = 9 \ imes 11 + 4 = 99 + 4 = 103 )", "Check: ( 103 \div 9 = 11 \ imes 9 = 99 ), remainder 4 → ( 103 \equiv 4 \pmod{9} ) ✅", "#### For ( x \equiv 7 \pmod{9} ):", "Find smallest ( k ) such that:\n( 9k + 7 \geq 100 ) → ( 9k \geq 93 ) → ( k \geq 10.\overline{3} )\nSo smallest integer ( k = 11 )", "Then:\n( x = 9 \ imes 11 + 7 = 99 + 7 = 106 )", "Check: ( 106 \div 9 = 11 \ imes 9 = 99 ), remainder 7 → ( 106 \equiv 7 \pmod{9} ) ✅", "---", "### Step 4: Compare and select the smallest", "We found:\n- Smallest ( x \equiv 4 \pmod{9} ), ≥ 100: 103\n- Smallest ( x \equiv 7 \pmod{9} ), ≥ 100: 106", "Thus, the smallest three-digit number congruent to either 4 or 7 modulo 9 is 103.", "---", "### Bonus Insight: Why 103?", "- 103 is the first three-digit number in the sequence of numbers ≡ 4 mod 9 starting from 103, 112, 121…\n- The sequence of numbers ≡ 4 mod 9 begins at ( 100 - (100 \mod 9) + 4 ) or adjusted upward.\n- Since 100 ≡ 1 mod 9, add ( (4 - 1) \mod 9 ) → 3 more → 103", "Similarly, for 7: 100 ≡ 1, so add 6 → 106", "---", "### Final Summary", "- Smallest three-digit number ≡ 4 mod 9: 103\n- Smallest three-digit number ≡ 7 mod 9: 106\n- Therefore, the smallest three-digit number congruent to 4 or 7 modulo 9 is 103", "Whether you're solving math problems, teaching modular arithmetic, or just curious, knowing how to efficiently determine such numbers saves time and deepens understanding.", "Key takeaway: Always compute the residue of 100 modulo 9, then adjust upward by the smallest amount to reach 4 or 7 — whichever gives the smallest valid three-digit value.", "---", "Keywords: smallest three-digit number congruent to 4 modulo 9, smallest three-digit number ≡ 7 mod 9, find x ≡ 4 or 7 (mod 9), modular arithmetic example, three-digit congruence problem, number theory guide", "Meta Description: Find the smallest three-digit number congruent to 4 or 7 modulo 9. Learn step-by-step how to solve this modular arithmetic problem with clear examples and practical insights.", "---", "Optimize your knowledge today — and next time you face a similar question, apply this quick method!"]
