What is the value of ((131)39+11⋅(−11)) mod 13?

Question

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
- Congruence arithmetic: if a ≡ b (mod m), then a^k ≡ b^k (mod m), and sums/products reduce modulo m at each step.
- Reduce large bases modulo m before exponentiation to simplify.
- The canonical remainder mod m is usually taken in 0..m−1.

2. Step-by-step reasoning
Reduce each part of the expression modulo 13 and then combine.
Expression: (131^39 + 11·(−11)) mod 13.

3. Intermediate calculations
- Reduce the base 131 modulo 13: 131 = 13·10 + 1, so 131 ≡ 1 (mod 13).
- Then 131^39 ≡ 1^39 ≡ 1 (mod 13).
- Compute 11·(−11) = −121.
- Reduce 121 modulo 13: 121 = 13·9 + 4, so 121 ≡ 4 (mod 13). Therefore −121 ≡ −4 (mod 13).
- Convert −4 to a standard nonnegative residue: −4 ≡ 13 − 4 = 9 (mod 13).
- Now combine: 131^39 + 11·(−11) ≡ 1 + (−121) ≡ 1 + 9 = 10 (mod 13).

4. Why the correct answer is correct
Because 131 reduces to 1 mod 13, its 39th power remains 1. The product 11·(−11) is −121 which reduces to 9 mod 13. Summing 1 and 9 gives the remainder 10 when divided by 13. All steps follow the rules of modular arithmetic.

5. Why the other options are incorrect
- 3: No intermediate step yields 3; incorrect reduction.
- 9: This is the reduction of −121 alone, but you must add the 131^39 term (which is 1), so 9 is incomplete.
- 23: Although 23 ≡ 10 (mod 13), the standard remainder in 0..12 is 10; the problem asks for the value mod 13, so 10 is the canonical representative (and the provided correct choice).

6. Conclusion
The computed remainder is 10, which matches the provided correct answer.

What is the value of ((131)39+11⋅(−11)) mod 13?单项选择题

类似问题

For integers a, b and k>0 which is NOT correct.

What is the value of: (2605321+4228) mod 13?

Select the correct value for 740 mod 2399. The following equalities may be useful: 72 mod 2399=49 74 mod 2399=2 78 mod 2399=4 716 mod 2399=16 732 mod 2399=256 764 mod 2399=512

What is the value of (260⋅5321+42⋅28) mod 13?

Use the fact that 7500 mod 26=3 to select the correct value for 71001 mod 26.

What is 14*4 in Z7 ( mod 7 - or wrapped around 7)?

Select the correct values for: 2916 mod 7.

Solve

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单