Number

Types of Numbers

Numbers can be grouped into different categories based on their properties. This helps us create definitions and rules for how different types of numbers behave. For example, we can define the factors of $N$ as pairs of integers that produce $N$ when multiplied together.

Common Factors

Factors that are shared by multiple numbers are called common factors.

$42$ and $69$ share $3$ as a factor
$18$ and $30$ have $2, 3, 6$ as common factors

The best way to find common factors is to use Pihedron's Prime Factor Table. By breaking down a number into a tree, we can not only solve for its prime factors but also multiply different combinations of these prime factors to obtain the full set of factors.

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The right column contains all the prime factors. $60$ has a total of $12$ factors which can be obtained by multiplying every combination of prime factors.

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The highest common factor (HCF) or greatest common denominator (GCD) of multiple numbers can be found by multiplying all their common prime factors together.

2, 2, 3, 3

2, 2, 3

HCF

2, 2, 3, 5

Common Multiples

Natural numbers have an infinite number of multiples. A multiple of $N$ is a natural number in the form $M \times N$ where $M$ is another natural number. We often see multiples in the form of times tables.

Most of the time, we are only concerned with the lowest common multiple (LCM) of multiple numbers. THe LCM of:

$2$ and $3$ is $6$
$6$ and $15$ is $30$

Note that the LCM of 2 numbers $A$ and $B$ is not always $A \times B$ . This is because the LCM of $A$ and $B$ is actually $A \times B \div C$ where $C$ is the HCF of $A$ and $B$ . The LCM can be thought of as the union of the prime factors of $A$ and $B$ .

Prime Factors Among Us

A = 5^2 \times 7^4 \times 11^p \newline B = 5^m \times 7^{n- 5} \times 11

Where $m$ , $n$ , and $p$ are integers such that:

m > 2 \newline n > 10 \newline p > 1

Fortunately, this question has given us the prime factors of both numbers. This means finding the highest common factor is as easy as plugging this question into a formula.

5^{\min(2, m)} \times 7^{\min(4, n - 5)} \times 11^{\min(p, 1)}

The highest common factor is found by raising each prime to its lowest exponent while the lowest common multiple is found by raising each prime to its highest exponent.

Since we are concerned with the minimum exponent, we want to know the minimum value of these variables to see if they will affect the highest common factor.

m_{min} = 3 \newline n_{min} = 11 \newline p_{min} = 2

When we substitute these lower bounds into the formula, it's obvious that the variables have no effect on the highest common factor.

5^{\min(2, 3)} \times 7^{\min(4, 6)} \times 11^{\min(2, 1)}

The final answer is straightforward.

5^2 \times 7^4 \times 11^1

4MA1 Jan 2022 1HR

How do we find all the pairs of numbers that have a certain HCF and LCM?

Sometimes it's possible to use Pihedron's Identity to solve problem's involving HCF and LCM. The following example has strong connections with the identity.

The Missing Pair

Two positive integers are each greater than 25.
Their lowest common multiple (LCM) is 216.
Their highest common factor (HCF) is 18.

Find the two integers.

\text{LCM}(A, B) = 216 \newline \text{HCF}(A, B) = 18

We can use Pihedron's Theorem to turn this into a counting problem. First, we divide the LCM by the HCF.

\frac{216}{18} = 12

Next, we count the number of distinct prime factors of $12$ .

12 = 2^2 \times 3^1 \newline \omega(12) = 2

$12$ has $2$ distinct prime factors. This means there are only $2^{2-1}=2$ possible pairs to consider. The first pair doesn't work because $A, B > 25$ is not satisfied.

A = 18 \newline B = 18 \times 2^2 \times 3^1

However, the second pair does.

A = 18 \times 2^2 \newline B = 18 \times 3^1

Hence, we get $72$ and $54$ as our answers.

4024 Winter 2021 Paper 1 Variant 2

Sets

Set theory has a lot of overlap with probability and boolean logic. You are expected to learn set notation to describe operations on sets.

But what even is a set?

A set is a collection of distinct items of the same type. The distinct trait means that sets do not contain any duplicates.

Before we dive into the fun stuff, it's important to know how we represent different set operations.

$\text{n}(A)$ represents the number of elements in set $A$
$x \in A$ means $x$ is an element of set $A$
$x \notin A$ means $x$ is not an element of set $A$
$A'$ is the complement of set $A$
$\varnothing$ is the empty set
$\mathscr{E}$ is the universal set
$A \subseteq B$ means $A$ is a subset of $B$
$A \nsubseteq B$ means $A$ is not a subset of $B$
$A \cup B$ is the union of sets $A$ and $B$
$A \cap B$ is the intersection of sets $A$ and $B$

Set Operations

There are 3 fundamental set operations.

$A'$ $A^{'}$ is the set that contains all the elements in $\mathscr{E}$ $E$ except those in set $A$ $A$
- acts like its opposite
- does not share any common elements with the original set
- $A \cap A' = \varnothing$
$A \cup B$ $A \cup B$ is the set that includes all the elements present in sets $A$ $A$ and $B$ $B$
- same as joining sets together to form a new set
- always the same size if not bigger than the individual sets
- $\text{n}(A \cup B) \ge \text{n}(A)$ and $\text{n}(A \cup B) \ge \text{n}(B)$
- always the same size if not smaller than the sum of the sizes of each set
- $\text{n}(A \cup B) \le \text{n}(A) + \text{n}(B)$
$A \cap B$ $A \cap B$ is the set that only contains the elements present in both sets $A$ $A$ and $B$ $B$
- always the same size if not smaller than the individual sets
- $\text{n}(A \cup B) \le \text{n}(A)$ and $\text{n}(A \cup B) \le \text{n}(B)$

Set Formulas

The sum of the sizes of each set equals the size of their union plus their intersection. This is because when you add the sizes of both sets, you are adding the size of their intersection twice.

\text{n}(A) + \text{n}(B) = \text{n}(A \cup B) + \text{n}(A \cap B)

The same idea can be applied to HCF and LCM.

A \times B = \text{LCM}(A, B) \times \text{HCF}(A, B)

Venn Diagrams

A Venn diagram is a visual representation of a group of sets. The rectangle represents the universal set while each circle represents a set.

Each enclosed area of a Venn diagram is an intersection of sets or their complements.

A \cap B'

A' \cap B

A \cap B

A' \cap B'

(A' \cup B')'

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