What is the difference between the HCF and LCM in terms of their prime factor powers?

The HCF is the product of the lowest powers of the common prime factors, while the LCM is the product of the highest powers of all prime factors present in the numbers.

How does the Venn diagram representation of HCF differ from LCM?

The HCF is the product of the prime factors in the intersection (overlap) of the circles, whereas the LCM is the product of all prime factors in the union (the entire area of both circles).

When finding the square root of a number using PFD, what must you do to the indices?

You must halve all the indices in the prime factor decomposition. This works because $(\text{base}^n)^{1/2} = \text{base}^{n/2}$.

What is a common mistake when calculating the LCM of two numbers from a list of their prime factors?

A common error is only multiplying the unique factors of each number and forgetting to include the shared factors, or conversely, multiplying the shared factors twice.

Why might a student incorrectly identify a number as a perfect square using PFD?

They might only check if the prime factors themselves are 'square-like' or if some indices are even, rather than ensuring that *every* single index in the decomposition is even.

What error occurs if you include a prime factor in the HCF that is not present in all numbers being compared?

The resulting HCF will be too large and will not be a divisor of the number that lacks that specific prime factor.

Define the 'Fundamental Theorem of Arithmetic'.

It states that every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers, regardless of the order.

What is the mathematical relationship between two numbers $a$ and $b$, their HCF, and their LCM?

The product of the two numbers is equal to the product of their HCF and LCM: $a \times b = HCF(a, b) \times LCM(a, b)$.

What condition must the indices of a PFD meet for a number to be a perfect cube?

Every index (exponent) in the prime factor decomposition must be a multiple of 3 (e.g., $3, 6, 9, \dots$).

Why does halving the indices of a PFD result in the square root of the number?

Squaring a number doubles its prime factor indices; therefore, the inverse operation (square root) requires halving those indices to find the original base factors.

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Uses of Prime Factor Decomposition

Summary

Prime Factor Decomposition (PFD) is the process of breaking down a composite number into a product of its prime building blocks. This unique 'mathematical signature' allows for the efficient calculation of common factors, multiples, and the identification of number properties like perfect squares and cubes.

1. Definition & Core Concepts

Prime Factor Decomposition (PFD) is the expression of a composite number as a product of prime numbers. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization, regardless of the order of the factors.
This process acts as the 'DNA' of a number, revealing its internal structure and all possible divisors. For example, if a number's PFD contains $2^3$ , we know it is divisible by $2$ , $4$ , and $8$ .
PFD is typically achieved using factor trees or repeated division by prime numbers until only primes remain.

2. Finding the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor, is the largest integer that divides two or more numbers without leaving a remainder.
Using PFD, the HCF is found by identifying the common prime factors shared by the numbers. If using index notation, the HCF is the product of the lowest power of each common prime factor.
In a Venn diagram approach, the HCF is the product of all prime factors located in the intersection (the overlapping middle section) of the sets representing each number.

Venn diagram showing the relationship between HCF (intersection) and LCM (union) using prime factors.

3. Finding the Lowest Common Multiple (LCM)

4. Identifying Perfect Squares and Cubes

5. Key Distinctions: HCF vs. LCM

6. Exam Strategy & Tips