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Zorluk: OrtaDivisibility Rules and Remainder Theorem

An industrial control system tracks part serial numbers using a cyclic validation algorithm. A batch identifier NN is generated using the formula N=65202−32145−45N = 65^{202} - 32^{145} - 45. The system validates this batch by computing the positive remainder when NN is divided by the base modulus 3333. What is the expected positive remainder for this batch?

Cevap: 23

Cevap

The expected positive remainder is 23.
By applying the principles of modular arithmetic, we can simplify each term of the expression 65202−32145−4565^{202} - 32^{145} - 45 modulo 3333. First, 65≡−1(mod33)65 \equiv -1 \pmod{33}, so 65202≡(−1)202=165^{202} \equiv (-1)^{202} = 1. Next, 32≡−1(mod33)32 \equiv -1 \pmod{33}, so 32145≡(−1)145=−132^{145} \equiv (-1)^{145} = -1. Substituting these simplified values back into the expression yields 1−(−1)−45=2−45=−431 - (-1) - 45 = 2 - 45 = -43. To find the positive remainder when dividing by 3333, we add a multiple of 3333 (in this case, 6666) to −43-43, which results in 2323.

Adım Adım Çözüm

1
Express the base numbers 6565 and 3232 modulo 3333.
65≡−1(mod33)65 \equiv -1 \pmod{33} and 32≡−1(mod33)32 \equiv -1 \pmod{33}
Finding remainders close to 11 or −1-1 significantly simplifies the evaluation of large exponents.
2
Evaluate the first term, 6520265^{202} modulo 3333.
(−1)202=1(-1)^{202} = 1
An even power of a negative one results in positive one.
3
Evaluate the second term, 3214532^{145} modulo 3333.
(−1)145=−1(-1)^{145} = -1
An odd power of a negative one results in negative one.
4
Substitute the simplified terms back into the original expression modulo 3333.
N≡1−(−1)−45≡2−45≡−43(mod33)N \equiv 1 - (-1) - 45 \equiv 2 - 45 \equiv -43 \pmod{33}
Combines all parts of the expression to find the overall remainder before final normalization.
5
Convert the negative remainder into a valid positive remainder.
−43≡−43+66=23(mod33)-43 \equiv -43 + 66 = 23 \pmod{33}
Remainders must be positive integers between 00 and the modulus minus one. Adding the smallest multiple of 3333 that is greater than 4343 (which is 6666) gives the valid positive remainder.

Anahtar Kavram

Modular Arithmetic and Remainder Theorem with Negative Remainders

Alternatif Yöntem

You could also simplify the constant term first: −45≡−12≡21(mod33)-45 \equiv -12 \equiv 21 \pmod{33}. Then the expression becomes 1−(−1)+21=231 - (-1) + 21 = 23.
Tahmini Süre:1m 30s
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Divisibility Rules and Remainder Theorem

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