# Difference between revisions of "Real Numbers"

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Concepts Lectures on Real Numbers and Divisibility | Concepts Lectures on Real Numbers and Divisibility | ||

− | =Concept Lectures= | + | ==Concept Lectures== |

Session # 1 [https://youtu.be/A7Xt-2CtzK4/ What are Real Numbers?] | Session # 1 [https://youtu.be/A7Xt-2CtzK4/ What are Real Numbers?] | ||

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[https://youtu.be/nvMBua1yNoE/ Problem Solving Session # 2] Show that the square of any positive integer is of the form of 3m or 3m+1 for some integer m. | [https://youtu.be/nvMBua1yNoE/ Problem Solving Session # 2] Show that the square of any positive integer is of the form of 3m or 3m+1 for some integer m. | ||

− | [https://youtu.be/Rk5-XKjq1-U/ Problem Solving Session # 3]Prove that one of every three consecutive positive integers is divisible by 3 | + | [https://youtu.be/Rk5-XKjq1-U/ Problem Solving Session # 3] Prove that one of every three consecutive positive integers is divisible by 3 |

+ | |||

+ | ==Concept Lectures== | ||

Session # 10 [https://youtu.be/VF4NtFu4Qgw/ Euclid's Division Algorithm] | Session # 10 [https://youtu.be/VF4NtFu4Qgw/ Euclid's Division Algorithm] | ||

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Session # 15 [https://youtu.be/sBGt9ncvd_I/ Expressing GCD of Two Positive Integers as a Linear Combination] | Session # 15 [https://youtu.be/sBGt9ncvd_I/ Expressing GCD of Two Positive Integers as a Linear Combination] | ||

+ | |||

+ | ==Problems== | ||

+ | |||

+ | [https://youtu.be/s-PcZWhRCjQ/ Problem Solving Session # 4] Find the GCD/HCF of 237 and 81 and express it as a linear combination of 237 and 81 | ||

+ | |||

+ | [https://youtu.be/6FwrDhqnjlU/ Problem Solving Session # 5] Find the GCD/HCF of 72 and 56 and express it as a linear combination of 72 and 56. Also show that the linear combination is not unique. | ||

+ | |||

+ | [https://youtu.be/E24re8lGdWc/ Problems Solving Session # 6] Word Problem | ||

+ | |||

+ | [https://youtu.be/E24re8lGdWc/ Problems Solving Session # 7] Word Problem | ||

+ | |||

+ | |||

+ | ==Concept Lectures== | ||

Session # 16 [https://youtu.be/MvslTd4_xgw/ Fundamental Theorem of Arithmetic] | Session # 16 [https://youtu.be/MvslTd4_xgw/ Fundamental Theorem of Arithmetic] | ||

− | = | + | ==Problems== |

− | [https://youtu.be/ | + | |

+ | [https://youtu.be/lbv3wZwGuuI/ Problems Solving Session # 8] Express 168 and 234 as a product of prime factors | ||

+ | |||

+ | [https://youtu.be/77wTxZg5pIM/ Problems Solving Session # 9] Prove that there is no natural number n for which 4^n ends with digit zero | ||

− | [https://youtu.be/ | + | [https://youtu.be/KsXrJ1RvrV4/ Problem Solving Session # 10] Prove that there are infinitely many prime numbers |

− | [https://youtu.be/ | + | [https://youtu.be/sAJyKNS3AXk/ Problem Solving Session # 11]Prove that a positive integer n is a prime number, if no prime less than or equal to square root of n, divides n |

## Revision as of 19:00, 16 March 2019

Concepts Lectures on Real Numbers and Divisibility

## Concept Lectures

Session # 1 What are Real Numbers?

Session # 2 What is meant by divisibility?

Session # 3 Properties of divisibility : Part 1

Session # 4 Properties of divisibility : Part 2

Session # 5 Euclid's Division Lemma : Part 1

Session # 6 Euclid's Division Lemma : Part 2

Session # 7 Euclid's Division Lemma : Part 3

Session # 8 Proof of the Euclid's Division Lemma

Session # 9 Euclid's Division Lemma - Applications

## Problems

Problem Solving Session # 1 Show that n^2-1 is divisible by 8 if n is an odd positive integer.

Problem Solving Session # 2 Show that the square of any positive integer is of the form of 3m or 3m+1 for some integer m.

Problem Solving Session # 3 Prove that one of every three consecutive positive integers is divisible by 3

## Concept Lectures

Session # 10 Euclid's Division Algorithm

Session # 11 Euclid's Division Algorithm: What is GCD?

Session # 12 Euclid's Division Algorithm: What are co-prime numbers?

Session # 13 Euclid's Division Algorithm: Finding GCD of Two Positive Integers

Session # 14 Euclid's Division Algorithm: Theorem 1 and its proof

Session # 15 Expressing GCD of Two Positive Integers as a Linear Combination

## Problems

Problem Solving Session # 4 Find the GCD/HCF of 237 and 81 and express it as a linear combination of 237 and 81

Problem Solving Session # 5 Find the GCD/HCF of 72 and 56 and express it as a linear combination of 72 and 56. Also show that the linear combination is not unique.

Problems Solving Session # 6 Word Problem

Problems Solving Session # 7 Word Problem

## Concept Lectures

Session # 16 Fundamental Theorem of Arithmetic

## Problems

Problems Solving Session # 8 Express 168 and 234 as a product of prime factors

Problems Solving Session # 9 Prove that there is no natural number n for which 4^n ends with digit zero

Problem Solving Session # 10 Prove that there are infinitely many prime numbers

Problem Solving Session # 11Prove that a positive integer n is a prime number, if no prime less than or equal to square root of n, divides n