# NPTEL Computational Number Theory and Algebra Assignment 1 Answer

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## NPTEL Computational Number Theory and Algebra Assignment

Algebra plays an important role in both finding algorithms, and understanding the limitations of computation. This course will focus on some of the fundamental algebraic concepts that arise in computation, and the algebraic algorithms that have applications in real life. The course will cover the problems of fast integer (or polynomial) multiplication (or factoring), fast matrix multiplication, primality testing, computing discrete logarithm, error-correcting codes, lattice- based cryptography, etc. The course intends to introduce both basic concepts and practical applications.

**INTENDED AUDIENCEÂ :Â**Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.

**Next Week Assignment Answers**

This course can have Associate in Nursing unproctored programming communication conjointly excluding the Proctored communication, please check announcement section for date and time. The programming communication can have a weightage of twenty fifth towards the ultimate score.

- Assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
**( All assignments in a particular week will be counted towards final scoring â quizzes and programming assignments).Â**- Unproctored programming exam score = 25% of the average scores obtained as part of Unproctored programming exam â out of 100
- Proctored Exam score =50% of the proctored certification exam score out of 100

**YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF ASSIGNMENT SCORE >=10/25 AND**

**UNPROCTORED PROGRAMMING EXAM SCORE >=10/25 AND PROCTORED EXAM SCORE >= 20/50.Â**

**If any one of the 3 criteria is not met, you will not be eligible for the certificate even if the Final score >= 40/100.Â**

**CHECK HERE OTHERS NPTEL ASSIGNMENTS ANSWERSÂ **

*BELOW YOU CAN GET YOUR NPTEL Computational Number Theory and Algebra Assignment 1 Answer 2022***?** :

**?**:

*1 point*

Which of the following pair of integers are coprime?

108, 45

1243,110

110,100

8,1

*1 point*

Which of the following equals gcd(108,48)?

gcd(48,1)

gcd(108,1)

gcd(48,12)

gcd(108,6)

ans – d

*1 point*

Which of the following is false?

nlogn=O(n2)nlogâĄn=O(n2)

n=o(n)n=o(n)

nlogn=Î©(n)nlogâĄn=Î©(n)

123456â n2=O(n2)123456â n2=O(n2)

ans –Â d

*1 point*

**Probabilistic algorithms are good?**

In lectures, you learned about the class BPPBPPÂ of randomized algorithms, which are efficient (i.e, run in time polynomial in the size of the input) but make errors with âlowâ probability (less than 1/2). Here, you will learn that randomized algorithms are âgoodâ (as asked in self-assessment). One reason is that they are often very simple in comparison to the deterministic algorithms. The other reason is, although they make errors, they are as good as (or even better than) their deterministic counterparts in practice.

You saw an example of a randomized algorithm in Assignment-0 which finds an even integer from an array AAÂ of 2n2nÂ consecutive integers in just one query with probability 1/2 while the best deterministic algorithm needs at least n+1n+1Â queries. Think about the situation when AAÂ has 22 million entries (around n=220n=220 ). Let us run the algorithm log2nlog2âĄnÂ times until we get the answer (i.e, just 2020Â queries). What is the probability that the algorithm gets at least one even number in log2nlog2âĄnÂ queries?

1â1n1â1n (around 0.99999990.9999999).

1n1n (around 0.00000010.0000001).

1313

1212

ans –Â d

*1 point*

In lectures, you have learned about some mathematical objects: rings, fields and domains. Based on this answer the following question.

Let,

Mn(R)Mn(R) be the set of all nĂnnĂnÂ matrices over reals with matrix addition and multiplication operations,

GLn(R)GLn(R) be the set of all nĂnnĂnÂ invertible matrices over reals with only matrix multiplication operation

and ZZÂ be the ring of all numbers of the form a+bâ5ââââa+bâ5, where aaÂ and bbÂ are integers.

Which of the following statements is/are true?

- Mn(R)Mn(R)is not a domain as it has zero-divisors.
- GLn(R)GLn(R)forms a group under matrix multiplication.
- Every number in ZZ has a unique factorisation.

Only 1.

Only 2.

Only 1 and 2.

All of 1, 2 and 3.

ans –Â d

*1 point*

You learned in lectures that an isomorphism is a one to one map that preserves the structure of two sets. Let ZZÂ be the set of integersÂ and EEÂ be the set of even integers. Which of the following is/are true?

- The map Ï:ZâEÏ:ZâEdefined as Ï(n)=2nÏ(n)=2nis an isomorphism.
- Any infinite cyclic group is isomorphic to (Z,+)(Z,+).

Only 1.

Only 2.

Both 1 and 2.

None of 1 and 2.

ans –Â d

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