P3923 University Mathematics Problem

Cirno: I got it! The answer must be 1! Rumia: What the heck (sweat). You should think again... I will give you the last problem first. This is a university mathematics problem.

Rumia: The Great Fairy wants to construct a finite field of order $ n $, where the elements are represented by the integers $ 0 \sim n - 1 $. A finite field must satisfy the following conditions: 1. There exists an additive identity $ o $, such that for any element $ a $, $ o + a = a + o = a $. 2. For any element $ a $, there exists an additive inverse $ a^{-1} $, such that $ a + a^{-1} = a^{-1} + a = o $. 3. There exists a multiplicative identity $ i $ different from the additive identity $ o $, such that for any element $ a $, $ i \times a = a \times i = a $. 4. For any element $ a $ that is not the additive identity, there exists a multiplicative inverse $ a^{-1} $, such that $ a \times a^{-1} = a^{-1} \times a = i $. 5. For any elements $ x $, $ y $, addition is commutative, i.e., $ x + y = y + x $. 6. For any elements $ x $, $ y $, multiplication is commutative, i.e., $ x \times y = y \times x $. 7. For any elements $ x $, $ y $, $ z $, addition is associative, i.e., $ ( x + y ) + z = x + ( y + z ) $. 8. For any elements $ x $, $ y $, $ z $, multiplication is associative, i.e., $ ( x \times y ) \times z = x \times ( y \times z ) $. 9. For any elements $ x $, $ y $, $ z $, multiplication distributes over addition, i.e., $ ( x + y ) \times z = x \times z + y \times z $. The Great Fairy certainly knows how to do it, but she wants to test you. In the output, the additive identity $ o $ is $ 0 $, and the multiplicative identity $ i $ is $ 1 $. # Description

A positive integer $ n $ ($ 2 \leq n \leq 350 $).

On the first line, output an integer $ k $. If a finite field of order $ n $ exists, then $ k = 0 $; otherwise $ k = -1 $. If $ k = 0 $, then: 1. Output an $ n $-by-$ n $ addition table of the finite field in the next $ n $ lines. The number in row $ i + 1 $, column $ j + 1 $ represents the result of $ i + j $ in the field. 2. Output an $ n $-by-$ n $ multiplication table of the finite field in the following $ n $ lines. The number in row $ i + 1 $, column $ j + 1 $ represents the result of $ i \times j $ in the field. In total, output $ n \times 2 + 1 $ lines. upd1: The SPJ is very strict. Do not output extra spaces at the ends of lines (the trailing newline at the end of the file will still be ignored). upd2: The official correct-answer file is large, so Luogu may keep judging... If this happens, please submit the source code directly.

| Test point | Range of $ n $ | Special property | | :-------: | :----------: | :-----------------: | | 1 | $ n = 3 $ | $ n $ is prime | | 2 | $ n = 4 $ | $ n $ is an integer power of $ 2 $ | | 3 | $ n = 6 $ | None | | 4 | $ n = 8 $ | $ n $ is an integer power of $ 2 $ | | 5 | $ n = 9 $ | None | | 6 | $ n = 19 $ | $ n $ is prime | | 7 | $ n = 89 $ | $ n $ is prime | | 8 | $ n = 181 $ | $ n $ is prime | | 9 | $ n = 233 $ | $ n $ is prime | | 10 | $ n = 25 $ | $ n $ is a square of a prime | | 11 | $ n = 121 $ | $ n $ is a square of a prime | | 12 | $ n = 169 $ | $ n $ is a square of a prime | | 13 | $ n = 27 $ | None | | 14 | $ n = 143 $ | None | | 15 | $ n = 128 $ | $ n $ is an integer power of $ 2 $ | | 16 | $ n = 81 $ | None | | 17 | $ n = 125 $ | None | | 18 | $ n = 243 $ | None | | 19 | $ n = 256 $ | $ n $ is an integer power of $ 2 $ | | 20 | $ n = 343 $ | None | Translated by ChatGPT 5