Kenzo works out that the HCF of two numbers is $$ 2^2\cdot3^2=\text{ }36.\text{ What two numbers might Kenzo have been using?} $$

Given that the Highest Common Factor (HCF) of two numbers is $ 2^2 \cdot 3^2 = 36 $, let's break down the problem to find potential pairs of numbers.

1. **HCF Breakdown:** The HCF tells us that both numbers share the prime factors $2^2$ and $3^2$. This means each number must include at least $2^2$ and $3^2$ in their prime factorizations.

2. **Prime Factor Constraints:** Each number must have a form that includes $2^2 \times 3^2 \times (\text{additional primes and/or higher powers})$. Let's express these numbers in general forms:
   \[
   a = 2^2 \times 3^2 \times m, \quad b = 2^2 \times 3^2 \times n,
   \]
   where $m$ and $n$ are coprime integers (they do not share any common prime factors). The HCF of $a$ and $b$ will then be $2^2 \times 3^2 = 36$.

3. **Examples of Possible Pairs:** To keep things simple, we can choose small values for $m$ and $n$. Here are a couple of examples:
   - If $m = 1$ and $n = 5$:
     \[
     a = 2^2 \times 3^2 \times 1 = 36, \quad b = 2^2 \times 3^2 \times 5 = 180.
     \]
     Therefore, one possible pair is $(36, 180)$.
   - If $m = 2$ and $n = 3$:
     \[
     a = 2^2 \times 3^2 \times 2 = 72, \quad b = 2^2 \times 3^2 \times 3 = 108.
     \]
     Another possible pair is $(72, 108)$.

4. **General Observation:** There are infinitely many pairs of numbers whose HCF is $36$. They can generally be expressed as:
   \[
   (2^2 \times 3^2 \times m, 2^2 \times 3^2 \times n),
   \]
   where $m$ and $n$ are coprime.

### Do you want more details or have any questions?

Here are 5 related questions that expand on this topic:
1. How would you find the HCF of more than two numbers?
2. If the HCF of two numbers is $36$ and their LCM is $1080$, what are the numbers?
3. What is the difference between the HCF and the least common multiple (LCM) of two numbers?
4. Can two numbers have an HCF greater than one if one of the numbers is a prime number?
5. How would the process change if the HCF was a different number, say $18$ instead of $36$?

**Tip:** To find possible pairs of numbers with a given HCF, ensure both numbers' prime factorizations include the given HCF, and explore different coprime multipliers for flexibility in choosing pairs.

Kenzo works out that the HCF of two numbers is 2^2 · 3^2 = 36. What two numbers might Kenzo have been using?

Explore how to find pairs of numbers with the HCF of 36 by breaking down their prime factorizations. Learn the concept of prime factors, the role of HCF, and how to apply the fundamental theorem of arithmetic to determine possible pairs of numbers. Suitable for Grades 8-10.

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