CNT101 Week 1

A number used for representing quantity and calculations.

The set of positive whole numbers starting from 1 and continuing infinitely.

The set of non-negative integers, including zero.

Numbers that are divisible by 2.

Numbers that are not divisible by 2.

Numbers whose factors are only one and itself, excluding 1.

Numbers with more than two factors.

The property that an integer number can be divided by another resulting in an integer number.

Numbers obtained by squaring a whole number. They are the result of multiplying a whole number by itself. Examples: 1, 4, 9, 16, 25...

Numbers obtained by cubing a whole number. They are the result of multiplying a whole number by itself thrice. Examples: 1, 8, 27, 64, 125...

A number that can be expressed as the product of two equal integers. For example, 9 is a square number because it is 3 multiplied by 3. (3 x 3 = 9).

A number that can be expressed as the product of three equal integers. For example, 27 is a cube number because it is 3 multiplied by 3 multiplied by 3 (3 x 3 x 3 = 27).

Prime numbers between 100 and 150 are 101, 103, 107, 109, 113, 127, 131, 137, 139, 149.

A prime number that can be expressed as the sum and difference of two other prime numbers. Example: 5 = 2 + 3, 5 = 7 - 2.

Common factors are numbers that divide into two or more numbers exactly. The common factors of 48, 54 and 72 are 1, 2, 3, and 6.

A sequence of three prime numbers. Examples: (2, 3, 5), (5, 7, 11), (17, 19, 23)

Sigma notation is a shorthand way of writing a sum of numbers. It uses the Greek letter sigma (∑) and specifies the starting and ending values of the index variable.

The product of consecutive integers starting from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

A perfect number is a positive integer that is equal to the sum of its proper divisors. A proper divisor is a divisor of a number excluding the number itself. For example, the proper divisors of 6 are 1, 2, and 3, and their sum is 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.

The difference between an even number and an odd number will always result in an odd number. This is because even numbers are divisible by 2, while odd numbers are not. When you subtract an odd number from an even number, you are essentially removing an odd quantity from a quantity that is divisible by 2. This leaves you with a quantity that is not divisible by 2, making the result an odd number.

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. An even number is any integer that is divisible by 2. The number 2 is the only even prime number because it is divisible by 1 and 2.

An even number can always be represented in the form of 2k, where k is any integer. This is because an even number is always divisible by 2. The general form 2k emphasizes this divisibility by 2.

In the Fibonacci sequence, two consecutive terms are always coprime. This means that the only common divisor of two consecutive terms is 1. For example, 8 and 13 are consecutive Fibonacci numbers and the only common divisor is 1.

If n is the square of an odd integer, then n can be represented in the form of (2k + 1)^2, where k is any integer. Expanding this expression, we get 4k^2 + 4k + 1, which can be rewritten as 4k(k + 1) + 1. Since k and k + 1 are consecutive integers, one of them must be even. Therefore, 4k(k + 1) is always divisible by 8, making the perfect square of the form 8k + 1.

If a number is divisible by 3, 4, and 5, it is also divisible by their least common multiple (LCM). The LCM of 3, 4, and 5 is 60. Therefore, any number divisible by 3, 4, and 5 is also divisible by 60.

If a is divisible by b, it means that b divides into a evenly, leaving no remainder. This can be mathematically expressed as a = bq, where q is an integer representing the quotient of the division.

The largest number that divides two or more integers without leaving a remainder. Example: The greatest common divisor (GCD) of 12 and 18 is 6.

The smallest number that is a common multiple of two or more given integers. Example: The least common multiple (LCM) of 12 and 18 is 36.

A method to find the greatest common divisor (GCD) of two integers by repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is zero. Example: The GCD of 12 and 18 is 6, found by dividing 18 by 12, then 12 by the remainder 6.

Expressing a number as a product of its prime factors. Example: 12 = 2 x 2 x 3.

A way of writing a large number with only its prime factors and their corresponding exponents. Example: 12 can be written as 2² x 3.

A shorthand way of writing a long addition of numbers following a specific pattern. Example: 5 + 10 + 15 + 20 can be written as ∑4𝑛=1 5𝑛 .

The base case is a specific instance of the statement that holds true, and it serves as the starting point for the induction proof. It usually involves checking the statement for the smallest value of n within the given range.

It involves assuming the statement is true for some arbitrary value of n (the inductive hypothesis) and then proving it to be true for the next value (n+1).

The binomial theorem provides a formula for expanding expressions of the form (x + y)n. It states that (x + y)n = ∑(n choose k) * x^(n-k) * y^k, where (n choose k) is the binomial coefficient, representing the number of ways to choose k items from a set of n items.

It involves multiplying the expression by itself n times. While conceptually simple, it becomes complex for larger values of n.

The coefficient of x^3 represents the numerical factor that multiplies the term x^3 after expanding the given expression.

It allows us to find the specific term with a given power of x (or y in this case) in an expanded expression without calculating all the terms.

The sum of consecutive squares from 1^2 to n^2 is equivalent to [n(n+1)(2n+1)] / 6

It's an assumption that the statement holds true for a specific value of n, usually n=1. This step helps to establish a starting point for the proof.