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

Question

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

BlackTom AI · Accepted Answer

7^2 = 49 shows the notation means exponentiation. Use modular exponentiation properties: (a·b) mod n = [(a mod n)·(b mod n)] mod n and a^(m+n) = a^m·a^n.

1. Relevant knowledge points and concepts
- Modular arithmetic: products and powers reduce modulo n at each step.
- Exponent decomposition: express the exponent as a sum of powers of two (binary exponentiation).
- Given equalities are residues of powers of 7 modulo 2399: 7^2 ≡ 49, 7^4 ≡ 2, 7^8 ≡ 4, 7^16 ≡ 16, 7^32 ≡ 256, 7^64 ≡ 512 (all mod 2399).

2. Complete reasoning step by step
- We need 7^40 mod 2399.
- Decompose 40 into powers of two: 40 = 32 + 8.
- Therefore 7^40 = 7^(32+8) = 7^32 · 7^8.
- Use the provided residues to get 7^32 mod 2399 and 7^8 mod 2399.

3. Intermediate calculations
- From given: 7^16 ≡ 16 (mod 2399). Then 7^32 = (7^16)^2 ≡ 16^2 = 256 (mod 2399) — matches provided 7^32 ≡ 256.
- From given: 7^4 ≡ 2 (mod 2399). Then 7^8 = (7^4)^2 ≡ 2^2 = 4 (mod 2399) — matches provided 7^8 ≡ 4.
- Multiply residues: 7^40 ≡ 7^32 · 7^8 ≡ 256 · 4 = 1024 (mod 2399).
- Since 1024 < 2399, the remainder is 1024.

4. Why the correct answer is correct
- The product of the known residues for exponents 32 and 8 gives 1024, and modular reduction does not change it because 1024 is already less than the modulus. This directly yields 7^40 ≡ 1024 (mod 2399).

5. Why other options are incorrect
- 260, 514, and 2048 are not equal to 256·4 = 1024, and none follow from the given residues via valid modular multiplications for exponent 40. They do not satisfy the relation 7^40 ≡ value (mod 2399) based on the provided powers.

6. Conclusion
The computed value 7^40 mod 2399 = 1024, which matches the provided correct answer: 1024.

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单项选择题

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