HCF/LCM - GCSE Maths

Name: HCF/LCM - GCSE Maths
Uploaded: 2024-05-28T21:34:16.000Z
Duration: 30 min 58 s

Understanding HCF and LCM

Introduction to HCF (Highest Common Factor)

The video begins by defining HCF as the highest common factor, which is a number that divides into another without leaving a remainder.

Factors of 12 are explored, showing pairs like (1, 12), (2, 6), and (3, 4) that multiply to give 12.

Common factors of two numbers are identified; for example, the common factors of 12 and 18 include 1, 2, 3, and 6.

Finding HCF Examples

The highest common factor of both numbers is determined; for instance, the HCF of 12 and 18 is found to be 6.

Another example with numbers 15 and 45 shows their factors leading to a common factor list: 1, 3, 5, and 15, with an HCF of 15.

The process extends to three numbers: finding the HCF of 16, 24, and 40 results in an HCF of eight after listing all factors.

Introduction to LCM (Lowest Common Multiple)

LCM is defined as the lowest common multiple; it involves identifying multiples from times tables.

An example using multiples of ten and twelve reveals that their first common multiple is sixty.

Finding LCM Examples

To find the LCM for four numbers—4, six, and nine—their respective multiples are listed until a common number appears.

The lowest number appearing in all lists is determined to be thirty-six.

Contextual Applications

A practical scenario involving Tom's haircuts every three weeks illustrates how to find when he will coincide with Mark's four-week schedule and Travis's five-week schedule again.

By continuing through their haircut schedules in multiples until they align again leads us to discover they will all get haircuts together in sixty weeks.

Additional Example on Distribution

Christine’s problem with distributing apples (18 total) into baskets highlights how maximum equal distribution can be achieved through understanding divisibility.

Understanding Factors and Baskets

Factors of 18 and 24

The number of baskets for apples must be a factor of 18, which includes: 1, 2, 3, 6, 9, and 18. This means we can use any of these numbers as the basket count.

For oranges, the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24. Thus the basket count for oranges must also be one of these factors.

Highest Common Factor (HCF)

To find the maximum number of baskets that can hold both fruits without leftovers requires identifying the highest common factor between the two sets: HCF(18) = 1,2,3,6 and HCF(24) = 1,2,3,4,6. The highest common factor is therefore 6.

With six baskets established as feasible for both fruits: each basket will contain 3 apples (from dividing total apples by baskets) and 4 oranges, leading to a total of 7 pieces of fruit per basket.

Finding HCF and LCM Using Prime Factorization

Introduction to Prime Factorization

When faced with larger numbers like 84 and 280, finding all factors manually can be tedious; thus prime factorization is recommended as an efficient alternative.

The prime factorizations are:

84 = 2^2 times 3 times 7

280 = 2^3 times 5 times 7. It’s often clearer to express these in expanded form rather than index form for visualization purposes.

Venn Diagram Method

A Venn diagram helps visualize common factors:

Place shared primes in the intersection (e.g., two '2's and one '7').

Unique primes go in their respective circles (e.g., '3' on left side for 84, '5' on right side for 280). This aids in calculating HCF easily by multiplying intersection values: HCF = (2 times 2 times 7) = 28.

Lowest Common Multiple (LCM)

To find LCM instead involves multiplying all unique primes from both sides of the Venn diagram:

Calculation yields LCM = (3 times (2^2) times (7) times (5)) = 840. This method efficiently determines multiples without exhaustive listing.

Using Calculators for Prime Factorization

Calculator Techniques

Demonstrated methods using both newer style calculators and older models to quickly derive prime factorizations.

For example:

Newer calculator shows 210 = (2 times 3 times5times7).

Older model reveals 735=(3times5times7^2). Utilizing calculators streamlines this process significantly when time is limited during exams or assessments.

Final Steps with Venn Diagrams

After obtaining prime factorizations via calculators:

Construct a new Venn diagram comparing 210 against 735.

Identify common factors ('3', '5', '7') placed centrally while unique ones ('2' only from 210) remain outside.

Finding Highest Common Factor and Lowest Common Multiple

Understanding the Process of HCF and LCM

The highest common factor (HCF) is calculated by multiplying all numbers found in the intersection of two sets. For example, for the numbers 3, 5, and 7, the HCF is 3 times 5 times 7 = 105.

To find the lowest common multiple (LCM), multiply all unique numbers from both sets. In this case, 2 times 3 times 5 times 7 times 7 = 1,470.

Prime Factorization Example

Given two prime factorized numbers A and B, we can directly use a Venn diagram to visualize their factors. For A: 2^8 times 5^4 times 11; for B: 2^3 times 5^1 times 11^3.

In analyzing both lists of factors:

Three instances of '2' are placed in the intersection.

One instance each of '5' and '11' also goes in the intersection.

Remaining factors include five '2's from A and three '5's from A; B contributes two more '11's.

Calculating HCF in Index Form

The HCF is determined by multiplying the intersecting factors: 2^3 cdot 5^1 cdot 11^1.

Since answers must be presented in index form, it simplifies to 2^3 cdot 5^1 cdot 11^1.

Calculating LCM in Index Form

The LCM involves multiplying all unique factors together. This results in an expression that can also be represented in index form as follows: