CNT101 Week 1

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Questions and Answers

What are natural numbers?

Numbers divisible by both 1 and itself.

Numbers that include fractions and decimals.

The set of non-negative integers including zero.

The set of positive whole numbers starting from 1. (correct)

Which of the following correctly describes whole numbers?

They start from zero and include all positive integers. (correct)

They can be negative numbers.

They include positive integers only.

They include both positive numbers and fractions.

Which type of numbers are divisible by 2?

Even numbers (correct)

Whole numbers

Odd numbers

Natural numbers

What defines prime numbers?

Numbers that can only be divided by 1 and itself. (D) Signup and view all the answers

Which best describes composite numbers?

Numbers with more than two factors. (A) Signup and view all the answers

What are integers?

The set of all whole numbers and their negative counterparts. (D) Signup and view all the answers

What property do numbers possess when they can be divided evenly by another integer?

They are divisible. (D) Signup and view all the answers

What is the definition of natural numbers, denoted by N?

Positive whole numbers starting from 1. (C) Signup and view all the answers

Which of the following is NOT a perfect number?

120 (A) Signup and view all the answers

The difference of an even number and an odd number is ____.

an odd number (A) Signup and view all the answers

Which of the following statements is true?

2 is an even prime number (B) Signup and view all the answers

What is the general form of an even number?

2k (B) Signup and view all the answers

If n, k ∈ ℤ such that n is the square of an odd integer, then perfect square must be of the form ____.

8k + 1 (A) Signup and view all the answers

If K is divisible by 3, 4 and 5, which of the following will also divide K?

3, 4, and 15 (A) Signup and view all the answers

If a is divisible by b, which of the following are true?

b is less than a (B) Signup and view all the answers

Which of the following is NOT a well-ordering set?

an open interval (0, 2) (A) Signup and view all the answers

Which of the following sequences represents the first five prime numbers?

2, 3, 5, 7, 11 (B) Signup and view all the answers

Which sequence correctly identifies the first five Fibonacci numbers?

0, 1, 1, 2, 3 (C) Signup and view all the answers

Identify the correct sequence of the first five square numbers.

1, 4, 9, 16, 25 (A) Signup and view all the answers

Which of the following describes even numbers?

Numbers divisible by 2 (C) Signup and view all the answers

In the context of number theory, which of these is an example of a composite number?

9 (D) Signup and view all the answers

Which sequence correctly lists the first five cube numbers?

1, 8, 27, 64, 125 (D) Signup and view all the answers

Identify which of the following sequences lists the first five natural numbers.

1, 2, 3, 4, 5 (D) Signup and view all the answers

What defines odd numbers in number theory?

Numbers that cannot be evenly divided by 2 (A) Signup and view all the answers

Which square numbers less than 500 are also cube numbers?

27 and 64 (C) Signup and view all the answers

How many prime numbers are there between 100 and 150?

10 (D) Signup and view all the answers

Which prime number can be expressed as the sum of two prime numbers?

11 (A) Signup and view all the answers

What are the common factors of 48, 54, and 72?

2, 3, 6 (A) Signup and view all the answers

How many triplets of prime numbers are there that are less than 300?

20 (A) Signup and view all the answers

Using sigma notation, how would you express the sum of the first 50 odd numbers?

$ ext{∑}_{i=1}^{50} (2i - 1)$ (D) Signup and view all the answers

What is the coefficient of the 7th term in the expansion of $(x + 3)^8$?

840 (C) Signup and view all the answers

Which binomial theorem expansion yields the term with a coefficient of 15?

$(2x - 1)^6$ (B) Signup and view all the answers

What is the expression for the sum of the series $rac{1}{2} imes rac{3}{2} + rac{1}{2} imes rac{5}{2} + rac{1}{2} imes rac{7}{2}$ when summed from $n=1$ to $n=20$?

64 (C) Signup and view all the answers

Which of the following represents the polynomial expansion of $(x - y)^5$?

x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5 (D) Signup and view all the answers

How does the Euclidean algorithm help in finding the greatest common divisor of two integers?

By repeatedly subtracting the smaller integer from the larger one. (B) Signup and view all the answers

Which of the following values is the least common multiple of 15 and 20?

60 (C) Signup and view all the answers

What is the prime factored form of 180?

2^2 imes 3^2 imes 5 (C) Signup and view all the answers

Which theorem provides a method to prove the infinitude of prime numbers?

Euclid's Theorem (A) Signup and view all the answers

What is the GCD of 48 and 180?

12 (D) Signup and view all the answers

What is the result of summing the first six terms of the series $(-1)^n$ from $n=1$ to $n=6$?

0 (B) Signup and view all the answers

Which of the following expressions is equivalent to the series $12 + 22 + 32 + … + n^2$?

$\frac{n(n + 1)(2n + 1)}{6}$ (B) Signup and view all the answers

What is the base case for the inequality $4n + 1 > (n+1)^2$ when $n = 2$?

$17 > 9$ (B) Signup and view all the answers

What is the coefficient of $x^3$ in the expansion of $(2x + 1)^{12}$?

1760 (C) Signup and view all the answers

Using the binomial theorem, what is the expanded form of $(x + 2y)^5$?

$x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5$ (A) Signup and view all the answers

What is the value of the sum $\sum_{n=1}^{100} (2n + 1)$?

10100 (D) Signup and view all the answers

What type of numbers does the term 'composite' refer to?

Whole numbers that are not prime (A) Signup and view all the answers

Which of the following is a characteristic of an even number?

It results in a remainder of 0 when divided by 2 (D) Signup and view all the answers

Which of the following statements accurately describes prime numbers?

They have exactly two distinct positive divisors. (A) Signup and view all the answers

Study Notes

Course Material

Number Theory is a branch of mathematics focused on the properties and relationships of whole numbers and integers.
This material covers fundamental concepts related to numbers and sequences.

Learning Outcomes

Students will learn the definitions and properties of fundamental number concepts and sequences.
They will use the Well-Ordering Property to prove propositions.
This material will explore sums and products of numbers, and related propositions.
Students will apply appropriate properties, and prove principles of Mathematical Induction, including applying those principles in solving problems related to integer equations, claims related to the Fibonacci sequence.
Students will understand and construct counterexamples in logic and mathematics, proving propositions on divisibility.
They will recognize and describe number patterns, using the Pascal's Triangle.

Resources Needed

Power Point Presentation, including topics like Number Concepts, Sequences, Well-Order Properties, Sum and Product, Fibonacci Sequence, Mathematical Induction, Binomial Coefficients, and Divisibility.
Video Links related to each topic.

Module Contents

Pretest (assessments of topics covered in previous module)
Unjumbled activity (reviewing concepts)
Number and Sequence Review
Sum and Product Review
Mathematical Induction Review
Binomial Coefficients Review
Divisibility Review
'Let's Go' Activities (application of topics)
Summaries
Key Terms
Post-test (assessment of entire module)
References

Number Concepts

Natural Numbers (N): Positive whole numbers starting from 1 (1, 2, 3, ...).
Whole Numbers (W): Non-negative integers (0, 1, 2, 3, ...).
Integers (Z): Positive and negative whole numbers, including zero (-3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers (Q): Numbers that can be expressed as a fraction where the numerator and denominator are integers (e.g., 1/2, 3/4).
Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
Real Numbers (R): The set of all rational and irrational numbers (-3, √2, π /2,....).

Types of Numbers

Even Numbers: Divisible by 2 (e.g., 2, 4, 6...).
Odd Numbers: Not divisible by 2 (e.g., 1, 3, 5...).
Prime Numbers: Positive integers greater than 1 with only 1 and itself as factors (e.g., 2, 3, 5, 7....).
Composite Numbers: Positive integers greater than 1 with factors other than 1 and itself (e.g., 4, 6, 8...).
Perfect Numbers: Numbers equal to the sum of their proper divisors.
Triangular Numbers: Numbers that can be arranged in equilateral triangles. (1, 3, 6, 10...).

Properties of Sums and Products

Distributive Property: a(b+c) = ab + ac.
Commutative Property: a + b = b + a and ab = ba.
Associative Property: (a + b) + c = a + (b + c) and (ab)c = a(bc).
Identity Property of Addition: a + 0 = a and 0 + a = a.
Identity Property of Multiplication: a * 1 = a and 1 * a = a.
Zero Property of Multiplication: a * 0 = 0 and 0 * a = 0.

Mathematical Induction

A proof technique for statements involving integers.

Fibonacci Sequence

A sequence of numbers in which each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5...).

Binomial Coefficients

Numbers that describe the combinations of choosing 'k' items from 'n' items, often used in binomial expansions. (C(n,k)).

Divisibility

A property of integers stating that a divides b, if and only if there exists an integer q such that aq=b.

Euclidean Algorithm

An algorithm for finding the greatest common divisor (GCD) of two integers.

The Fundamental Theorem of Arithmetic

Every integer greater than 1 can be represented uniquely as a product of primes.

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Description

This quiz covers essential concepts in Number Theory, focusing on properties and relationships of whole numbers and sequences. Students will explore the Well-Ordering Property, Mathematical Induction, and identify number patterns through resources like Pascal's Triangle. Join to test your understanding of these foundational topics!