Q.Let $A = \{9, 10, 11, 12, 13\}$ and let $f : A \to \mathbb{N}$ be defined by $f(n) =$ the highest prime factor of $n$. Find the range of $f$.

Prime Factor Mapping: Seeing Numbers as Their Building Blocks

Think of a number like 60. You know it breaks down as $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$. That's its prime factorisation — the unique set of prime numbers that multiply to give it.

Now imagine you could take that factorisation and turn it into a map — a kind of address or fingerprint that tells you everything about the number's multiplicative identity. That's the core idea of Prime Factor Mapping.

The Intuition

Every positive integer greater than 1 can be written as a product of primes, and this representation is unique (Fundamental Theorem of Arithmetic). A Prime Factor Map is simply a way to organise that information so you can compare numbers, find their GCD or LCM, or see relationships at a glance.

Think of it like a frequency table for primes. For 60, the prime 2 appears twice, 3 appears once, 5 appears once. For 84 = $2^2 \times 3 \times 7$, the map would show: 2 appears twice, 3 once, 7 once.

The Precise Statement

Let $n$ be a positive integer greater than 1. Its Prime Factor Map is the set

In simpler terms: for each prime that divides $n$, record the highest exponent such that $p^e$ still divides $n$.

For $n = 1$, the map is empty (no primes).

Why This Matters

The real power of Prime Factor Mapping is how it simplifies operations:

• GCD: Take the minimum exponent for each common prime.
• LCM: Take the maximum exponent for each prime that appears in either number.
• Divisibility: $a \mid b$ if and only if every prime in $a$'s map has an exponent $\leq$ its exponent in $b$'s map.

Example Walkthrough

Find GCD(60, 84) and LCM(60, 84) using Prime Factor Mapping.

Step 1: Write the maps

• 60: $\{ (2,2), (3,1), (5,1) \}$
• 84: $\{ (2,2), (3,1), (7,1) \}$

Step 2: GCD — take minimum exponents for common primes

Common primes: 2 and 3. Minimum exponent for 2 is 2, for 3 is 1. So:

Step 3: LCM — take maximum exponents for all primes that appear

Primes: 2 (max exponent 2), 3 (max 1), 5 (max 1), 7 (max 1). So: …