Alternatively, maybe it meant a number such that $ x + 3 $ is divisible by **each** of $ 7, 8, 9 $ individually, but not necessarily simultaneously. But that’s redundant — divisibility by each implies divisibility by their LCM.

["Understanding Divisibility by 7, 8, and 9: What It Truly Means", "When we encounter a problem asking for a number ( x ) such that ( x + 3 ) is divisible by each of 7, 8, and 9 individually, it’s natural to assume we’re searching for values satisfying multiple congruences simultaneously. But a key insight in number theory simplifies this: divisibility by 7, 8, and 9—each on its own—does not require finding a single number satisfying all at once in complex ways. Instead, because 7, 8, and 9 are pairwise coprime (except for shared factors between 8 and 9 being only 1), the condition frequently reduces to finding ( x + 3 ) divisible by their least common multiple (LCM).", "---", "### The Core Principle: LCM and Multiple Divisibility", "Let’s formalize:\nIf ( x + 3 ) is divisible by 7, 8, and 9 individually, then ( x + 3 ) must be divisible by\n[\n\ ext{LCM}(7, 8, 9)\n]\nThis is a fundamental concept: a number divisible by multiple pairwise coprime (or relatively prime) divisors must be divisible by their product (or LCM). Since 7, 8, and 9 have no common factors beyond 1:", "- Prime factorization:\n - ( 7 = 7 )\n - ( 8 = 2^3 )\n - ( 9 = 3^2 )", "Because all prime bases are distinct,\n[\n\ ext{LCM}(7, 8, 9) = 2^3 \ imes 3^2 \ imes 7 = 8 \ imes 9 \ imes 7 = 504\n]", "Thus, ( x + 3 ) must be divisible by 504 — that is,\n[\nx + 3 \equiv 0 \pmod{504}\n\quad \Rightarrow \quad\nx \equiv -3 \pmod{504}\n]", "---", "### Why This Is More Than Just “Divisible Individually”", "Many might assume divisibility by 7, 8, and 9 separately implies satisfying each condition independently without a uniform modular structure. But in fact:", "- Divisibility by each implies ( x \equiv -3 \pmod{7} )\n- ( x \equiv -3 \pmod{8} )\n- ( x \equiv -3 \pmod{9} )", "Since 7, 8, and 9 are relatively prime in pairs (no overlapping prime factors), the combined condition collapses cleanly to\n[\nx \equiv -3 \pmod{\ ext{LCM}(7,8,9)} = \pmod{504}\n]", "This modular single condition unifies all three divisibility checks. There's no need for separate cases or checks — being divisible by each is mathematically equivalent to being divisible by their LCM.", "---", "### Practical Solutions", "Any number of the form\n[\nx = 504k - 3 \quad \ ext{for integer } k\n]\nsatisfies the requirement:\n- ( x + 3 = 504k ), divisible by 7, 8, and 9.", "For example:\n- When ( k = 1 ), ( x = 501 ):\n ( 501 + 3 = 504 ), divisible by 7 ✅, 8 ✅, 9 ✅\n- When ( k = 2 ), ( x = 1005 ):\n ( 1005 + 3 = 1008 = 504 \ imes 2 ), also divisible by all three.", "This pattern holds for all integer ( k ).", "---", "### When Does It Become More Complex?", "If the problem explicitly required divisibility by, say, ( \ ext{lcm}(7, 8, 9, 10) ) or included a non-coprime divisor, then separate checks would apply. But with only 7, 8, and 9, the LCM fully encapsulates all required divisibility neatly.", "---", "### Summary", "- ( x + 3 ) divisible individually by 7, 8, and 9 ⇒ ( x + 3 ) divisible by ( \ ext{LCM}(7,8,9) = 504 )\n- This unifies checks into a single modular condition: ( x \equiv -3 \pmod{504} )\n- Solutions: ( x = 504k - 3 ), ( k \in \mathbb{Z} )", "Thus, asking whether ( x + 3 ) is divisible by each of 7, 8, and 9—without requiring individual compliance—trivially reduces to checking divisibility by their least common multiple. This insight avoids unnecessary complexity and leverages core number theory principles for clarity and efficiency.", "---", "Keywords:\ndivisible by 7, divisible by 8, divisible by 9, LCM of 7, 8, 9, x ≡ -3 mod 504, modular arithmetic, number theory, divisibility conditions.", "Meta description:\nDiscover why ( x + 3 ) divisible by 7, 8, and 9 individually reduces to divisibility by their LCM = 504. Learn how number theory simplifies multi-divisibility checks using modular arithmetic."]