NPTEL Computational Number Theory and Algebra Assignment 1 Answer

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NPTEL Computational Number Theory and Algebra Assignment

ABOUT THE COURSE :
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 AUDIENCE : Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.

PREREQUISITES : Preferable (but not necessary)– Theory of Computation, Algorithms, Algebra

INDUSTRIES SUPPORT : Cryptography, Coding theory, Computer Algebra, Symbolic Computing Software, Cyber Security, Learning Software

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.

Final score = Assignment score + Unproctored programming exam score + Proctored Exam 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 – quizzes and programming assignments).
Unproctored programming exam score = 25% of the average scores obtained as part of Unproctored programming exam – 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.

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.

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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?

$n \log n = O (n^{2})$

$n = o (n)$

$n \log n = Ω (n)$

$123456 \cdot n^{2} = O (n^{2})$

ans – d

1 point

Probabilistic algorithms are good?

In lectures, you learned about the class $B P P$ of randomized algorithms, which are efficient (i.e, run in time polynomial in the size of the input) but make errors with ‘low’ probability (less than 1/2). Here, you will learn that randomized algorithms are ‘good’ (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 $A$ of $2 n$ consecutive integers in just one query with probability 1/2 while the best deterministic algorithm needs at least $n + 1$ queries. Think about the situation when $A$ has $2$ million entries (around $n = 2^{20}$ ). Let us run the algorithm $\log_{2} n$ times until we get the answer (i.e, just $20$ queries). What is the probability that the algorithm gets at least one even number in $\log_{2} n$ queries?

$1 - \frac{1}{n}$ (around $0.9999999$ ).

$\frac{1}{n}$ (around $0.0000001$ ).

$\frac{1}{3}$

$\frac{1}{2}$

ans – d

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1 point

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

Let,

$M_{n} (R)$ be the set of all $n \times n$ matrices over reals with matrix addition and multiplication operations,

$G L_{n} (R)$ be the set of all $n \times n$ invertible matrices over reals with only matrix multiplication operation

and $Z$ be the ring of all numbers of the form $a + b \sqrt{- 5}$ , where $a$ and $b$ are integers.

Which of the following statements is/are true?

$M_{n} (R)$ is not a domain as it has zero-divisors.
$G L_{n} (R)$ forms a group under matrix multiplication.
Every number in $Z$ has a unique factorisation.

Only 1.

Only 2.

Only 1 and 2.

All of 1, 2 and 3.

ans – d

1 point

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

The map $φ : Z \to E$ defined as $φ (n) = 2 n$ is an isomorphism.
Any infinite cyclic group is isomorphic to $(Z, +)$ .

Only 1.

Only 2.

Both 1 and 2.

None of 1 and 2.

ans – d

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NPTEL Computational Number Theory and Algebra Assignment

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