In mathematics, the versatility of numbers is paramount. This chapter focuses on the dexterity of number conversions, the intricacies of prime factorisation, and the methodologies for deducing the Highest Common Factor (HCF) and Lowest Common Multiple (LCM). These concepts are not only vital for IGCSE students but also form the basis for advanced mathematical applications.

Number Conversions

Number conversions bridge the gap between numerical and verbal representations, playing a vital role in understanding mathematical concepts.

Converting Numbers to Words

• Single Digits (1-9): Each digit corresponds to a unique word, e.g., 1 to 'one'.
• Tens (10, 20, ... 90): Distinct words represent these multiples of ten, like 30 to 'thirty'.
• Hundreds and Thousands: Numbers in hundreds and thousands are expressed by combining the unit's word with 'hundred' or 'thousand', respectively. For example, 1,234 translates to 'one thousand two hundred and thirty-four'.

Converting Words to Numbers

• Decoding Units: Identify and translate words like 'thousand' or 'million' to their numerical values.
• Summing Values: Add the numerical values represented by each word to form the complete number.

Example: Word to Number Conversion

Convert 'two thousand and twenty-four' to a number. Solution:

• 'Two thousand' is interpreted as 2000.
• 'Twenty' corresponds to 20.
• 'Four' is directly converted to 4.
• Combining these gives 2000 + 20 + 4 = 2024.

Prime Factorisation

Prime factorisation breaks down a composite number into a product of its prime factors.

Image courtesy of Cuemath

Example: Prime Factorisation of 60

Solution:

1. Begin with 2: 60 is divisible by 2, giving 60 ÷ 2 = 30.

2. Continue with 2: Again, 30 is divisible by 2, leading to 30 ÷ 2 = 15.

3. Proceed to the next prime, 3: 15 is divisible by 3, yielding 15 ÷ 3 = 5.

4. Since 5 is a prime number, we stop here.

5. The prime factors of 60 are therefore: 2, 2, 3, 5 (or 2 x 2 x 3 x 5).

Highest Common Factor (HCF)

The HCF is the largest number that divides two or more numbers completely.

Methods

• Prime Factorisation Method: Find the common prime factors of the numbers and multiply them.
• Euclidean Algorithm: Repeatedly divide the larger number by the smaller and use the remainder in subsequent steps until it reaches zero.

Example: HCF of 12 and 18

Solution:

1. Prime factors of 12: 2 x 2 x 3.

2. Prime factors of 18: 2 x 3 x 3.

3. The common factors are one 2 and one 3. Therefore, the HCF is calculated by multiplying these common factors:

• HCF: $2 \times 3$ (since both 12 and 18 have one 2 and one 3 in their prime factors)
• Therefore, the HCF of 12 and 18 is $6$.

Lowest Common Multiple (LCM)

LCM is the smallest number that is a common multiple of two or more numbers.

Methods

• Prime Factorisation Method: Use the prime factors of the numbers, incorporating the highest power of each prime appearing in the factorisation of either number.
• Listing Multiples: Write the multiples of each number until a common value is found, but this can be inefficient for larger numbers.

Example: LCM of 100 and 200

Solution:

1. Prime factors of 100: $2 \times 2 \times 5 \times 5.$

2. Prime factors of 200: $2 \times 2 \times 2 \times 5 \times 5.$

3. For the Least Common Multiple (LCM), you take the highest power of each prime factor found in either number. In this case:

• For prime factor 2: the highest power is $2^3$ (from 200).
• For prime factor 5: the highest power is $5^2$ (both 100 and 200 have $5^2$).

4. Thus, the LCM of 100 and 200 is $2^3 \times 5^2 = 8 \times 25 = 200$.