But 504 is large. So no two-digit number satisfies $ x \equiv -3 \pmod{504} $.
["Understanding Why No Two-Digit Number Fulfills the Congruence $ x \equiv -3 \pmod{504} $", "When solving modular arithmetic problems, especially those involving congruences modulo a large number, it's common to wonder about the range of solutions. A particularly interesting case is when a two-digit number $ x $ is said to satisfy $ x \equiv -3 \pmod{504} $. But can any two-digit number actually satisfy this congruence? The answer is no — and understanding why involves exploring the nature of modular equivalence and the size of modulus 504.", "### What Does $ x \equiv -3 \pmod{504} $ Mean?", "The expression $ x \equiv -3 \pmod{504} $ means that when $ x $ is divided by 504, the remainder is equivalent to $ -3 $ modulo 504. Since remainders are conventionally non-negative, $ -3 \mod 504 $ is equivalent to $ 504 - 3 = 501 $. Therefore:", "[\nx \equiv -3 \pmod{504} \quad \Longleftrightarrow \quad x \equiv 501 \pmod{504}\n]", "This means all solutions for $ x $ are of the form:", "[\nx = 504k + 501 \quad \ ext{for integers } k\n]", "### Why No Two-Digit Number Fits This Form", "Two-digit numbers range from 10 to 99 inclusive. We now check whether any such number satisfies $ x = 504k + 501 $.", "- For $ k = 0 $: $ x = 504(0) + 501 = 501 $ — too large (3 digits).\n- For $ k = -1 $: $ x = 504(-1) + 501 = -504 + 501 = -3 $ — negative, invalid since we want positive two-digit numbers.", "No integer $ k $ produces a value between 10 and 99. Thus, no two-digit number satisfies $ x \equiv -3 \pmod{504} $.", "### Practical Implications", "This result shows how modular arithmetic with a large modulus restricts the number of feasible solutions within bounded ranges. While solutions exist (all congruent to 501 mod 504), they all exceed the two-digit range. In real-world applications—such as clock arithmetic, cryptography, or digital systems—this distinction matters: only a unique residue modulo a modulus can represent values within a given interval. Since 501 is far outside the two-digit domain, it cannot represent any valid two-digit number.", "### Summary", "- $ x \equiv -3 \pmod{504} $ is logically equivalent to $ x \equiv 501 \pmod{504} $.\n- The smallest non-negative solution is $ x = 501 $, which is a 3-digit number.\n- No two-digit number satisfies the congruence.", "Understanding modular constraints helps avoid apparent but impossible solutions, clarifying the intersection of number theory and practical number ranges.", "---", "Key Takeaway: When solving $ x \equiv -3 \pmod{504} $, the smallest solution is 501 — so no two-digit integer can satisfy it. This highlights how modular congruences, combined with modulus size, determine the range of viable solutions."]
