Today, 01:16 AM
An Interesting Number
Summary
This Stack Exchange discussion seeks a natural number $m$ whose four-digit square ($m^2 = ABCD$) and six-digit cube ($m^3 = EFGHIJ$) together contain all ten digits from $0$ to $9$ exactly once without repetition. By establishing size bounds ($47 \le m \le 98$), eliminating terminal digits that repeat upon exponentiation, and using modular arithmetic (noting that the sum of the digits $0$ through $9$ is $45$, meaning $m^2 + m^3$ must be divisible by $9$), the post narrows the search down to a handful of candidate numbers. Testing these candidates reveals that the unique solution is $m = 69$, whose square $69^2 = 4761$ and cube $69^3 = 328509$ combine to span every digit from $0$ to $9$.
ARTICLE
Summary
This Stack Exchange discussion seeks a natural number $m$ whose four-digit square ($m^2 = ABCD$) and six-digit cube ($m^3 = EFGHIJ$) together contain all ten digits from $0$ to $9$ exactly once without repetition. By establishing size bounds ($47 \le m \le 98$), eliminating terminal digits that repeat upon exponentiation, and using modular arithmetic (noting that the sum of the digits $0$ through $9$ is $45$, meaning $m^2 + m^3$ must be divisible by $9$), the post narrows the search down to a handful of candidate numbers. Testing these candidates reveals that the unique solution is $m = 69$, whose square $69^2 = 4761$ and cube $69^3 = 328509$ combine to span every digit from $0$ to $9$.
ARTICLE
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