Prime Factorisation Factors and Multiples for the TMUA

Multiples and Common Multiples

A multiple of a number is a value found in the times table of that specific number. A common multiple for two or more numbers is a value that is a multiple of all of those numbers. The Lowest Common Multiple (LCM) is the smallest positive number that is divisible by all the numbers in a given set.

Example: Finding common multiples

Find the first three common multiples of $6$ and $8$.

1. Multiples of $6$: $6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72...$
2. Multiples of $8$: $8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96...$
3. Common multiples: $24, 48, 72...$

Factors and Common Factors

A factor (also known as a divisor) of a number is a value that divides into that number exactly, leaving no remainder. A common factor is a number that divides exactly into all the numbers in a set. The Highest Common Factor (HCF) is the largest number that divides exactly into all the numbers provided.

Example: Finding common factors of two or more numbers

Find all the common factors of $12$ and $18$.

1. Factors of $12$: $1, 2, 3, 4, 6, 12$
2. Factors of $18$: $1, 2, 3, 6, 9, 18$
3. Common factors: $1, 2, 3, 6$

Prime Numbers and Factorisation

Prime numbers are integers that possess exactly two factors: $1$ and the number itself. Prime factorisation is the process of writing a number as a product of its prime factors. The Unique Factorisation Theorem (Fundamental Theorem of Arithmetic) guarantees that every integer greater than $1$ has a unique prime factorisation, regardless of the order of the factors.

Divisibility Tests

Divisibility tests are useful for identifying factors quickly:

1. 2: The last digit is even.
2. 3: The sum of the digits is divisible by $3$.
3. 4: The last two digits form a number divisible by $4$.
4. 5: The last digit is $0$ or $5$.
5. 6: The number is divisible by both $2$ and $3$.
6. 7: Subtract two times the last digit from the rest of the number: if the result is divisible by $7$, the original is too. For example, for $546$, calculate $54 - (6 \times 2) = 42$. Since $42$ is a multiple of $7$, $546$ is also a multiple of $7$.
7. 8: The last three digits form a number divisible by $8$.
8. 9: The sum of the digits is divisible by $9$.

Example: Deciding if a number is prime

Is $153$ a prime number? Sum of digits: $1 + 5 + 3 = 9$. Since $9$ is divisible by $3$, $153$ is divisible by $3$. It has more than two factors (at least $1, 3, 153$), so it is not prime.

Important Prime Facts

1. There are 10 primes less than 30: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
2. The number $1$ is not prime as it only has one factor.
3. The number $2$ is the only even prime.
4. Apart from $2$ and $5$, all primes end in $1, 3, 7,$ or $9$.

Prime Factorisation Methods

Example: Writing prime factorisation in index form

Write $180$ as a product of prime factors using index notation.

Method 1: Repeated Division (Ladder)

Divide $180$ by the smallest prime $2$ to get $90$. Divide $90$ by $2$ to get $45$. Divide $45$ by $3$ to get $15$. Divide $15$ by $3$ to get $5$. Stop as $5$ is prime. $180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5$.

Method 2: Factor Tree

As shown in the diagram, circle prime numbers at the end of each branch. The product of the circled numbers is $2^2 \times 3^2 \times 5$.

Finding HCF and LCM Using Prime Factorisation

Example: Finding the HCF and LCM of 180 and 420

1. Prime factorisations: $180 = 2 \times 2 \times 3 \times 3 \times 5$ $420 = 2 \times 2 \times 3 \times 5 \times 7$

2. Use a Venn diagram to find the overlap:

3. HCF: Multiply the numbers in the overlap: $2 \times 2 \times 3 \times 5 = 60$.

4. LCM: Multiply all numbers in the diagram: $2 \times 2 \times 3 \times 3 \times 5 \times 7 = 1260$.

Check: $HCF \times LCM = 60 \times 1260 = 75,600$. Also $180 \times 420 = 75,600$. The results match.

Example: HCF and LCM of three numbers (180, 168, and 72)

1. $180 = \underline{2} \times \underline{2} \times \underline{3} \times 3 \times 5$
2. $168 = \underline{2} \times \underline{2} \times \underline{3} \times 7$
3. $72 = \underline{2} \times \underline{2} \times 2 \times \underline{3} \times 3$

HCF: Find prime factors common to all: $2 \times 2 \times 3 = 12$. LCM: Include the maximum occurrence of each prime factor. The max number of $2$s is three ($2^3$), $3$s is two ($3^2$), and one $5$ and one $7$. $LCM = 2^3 \times 3^2 \times 5 \times 7 = 2520$.

Solving Problems and Finding Square Roots

Example: Factor verification

Show that $210$ is a factor of $3780$. $210 = 2 \times 3 \times 5 \times 7$. $3780 = 2^2 \times 3^3 \times 5 \times 7 = (2 \times 3 \times 5 \times 7) \times (2 \times 3^2) = 210 \times 18$. Since all prime factors of $210$ are present in the factorisation of $3780$, it is a factor.

Example: Solving logic problems with primes

Sarah reverses the digits of a 2 digit number and subtracts it from the original. Can the answer be prime? Let the original number be $10y + x$. The reversed is $10x + y$. The difference is $10y + x - (10x + y) = 9y - 9x = 9(y - x)$. Since this is a multiple of $9$, it cannot be prime (unless it is $9$, but $9$ is not prime).

Example: Finding square roots

Given $129,600 = 2^6 \times 3^4 \times 5^2$, find $\sqrt{129,600}$. Halve the indices: $2^{6/2} \times 3^{4/2} \times 5^{2/2} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$.