Question

1.8 factors, highest common factor (hcf)

the meaning of te arawa

some accounts say that te arawa was the largest waka hourua that came to aotearoa from hawaiki.
to find out the meaning of the name of te arawa, find the highest common factor (hcf) of the given numbers. the letter beside each question and its answer will give the puzzle code.

• 14, 35 & 56 (a)
• 20 & 35 (k)
• 18 & 36 (h)
• 63 & 84 (f)
• 18 & 45 (e)
• 60 & 90 (r)
• 16, 24 & 40 (o)
• 13, 39 & 52 (n)
• 15 & 45 (t)
• 32 & 48 (a)
• 12, 16 & 20 (s)
• 12, 36 & 48 (e)
• 45 & 90 (d)
• 48 & 72 (m)
• 20 & 60 (a)
• 42 & 70 (f)
• 18 & 24 (h)
• 33, 55 & 77 (i)
• 19 & 57 (k)
• 17 & 51 (o)
• 25, 75 & 100 (n)
• 26 & 52 (a)

Explanation:

🆕 New Concept Discovered: Highest Common Factor (HCF)
The largest number that divides two or more numbers exactly.

Step 1: Calculate the HCF for each lettered box

We find the highest common factor (HCF) for each set of numbers given in the puzzle:

* **A** (top left): \(14, 35 \text{ \& } 56\)
* Factors of \(14\): \(1, 2, 7, 14\)
* Factors of \(35\): \(1, 5, 7, 35\)
* Factors of \(56\): \(1, 2, 4, 7, 8, 14, 28, 56\)
* \(\text{HCF} = 7\)

* **K** (top middle-left): \(20 \text{ \& } 35\)
* \(\text{HCF} = 5\)

* **H** (top middle-right): \(18 \text{ \& } 36\)
* \(\text{HCF} = 18\)

* **F** (top right): \(63 \text{ \& } 84\)
* \(\text{HCF} = 21\)

* **E** (upper left): \(18 \text{ \& } 45\)
* \(\text{HCF} = 9\)

* **R** (upper middle): \(60 \text{ \& } 90\)
* \(\text{HCF} = 30\)

* **O** (upper middle-right): \(16, 24 \text{ \& } 40\)
* \(\text{HCF} = 8\)

* **N** (upper right): \(13, 39 \text{ \& } 52\)
* \(\text{HCF} = 13\)

* **T** (middle left): \(15 \text{ \& } 45\)
* \(\text{HCF} = 15\)

* **A** (center left): \(32 \text{ \& } 48\)
* \(\text{HCF} = 16\)

* **S** (center right): \(12, 16 \text{ \& } 20\)
* \(\text{HCF} = 4\)

* **E** (middle right): \(12, 36 \text{ \& } 48\)
* \(\text{HCF} = 12\)

* **D** (lower-middle left): \(45 \text{ \& } 90\)
* \(\text{HCF} = 45\)

* **M** (center bottom-left): \(48 \text{ \& } 72\)
* \(\text{HCF} = 24\)

* **A** (center bottom-right): \(20 \text{ \& } 60\)
* \(\text{HCF} = 20\)

* **F** (lower-middle right): \(42 \text{ \& } 70\)
* \(\text{HCF} = 14\)

* **H** (bottom left): \(18 \text{ \& } 24\)
* \(\text{HCF} = 6\)

* **I** (bottom middle-left): \(33, 55 \text{ \& } 77\)
* \(\text{HCF} = 11\)

* **K** (bottom middle-right): \(19 \text{ \& } 57\)
* \(\text{HCF} = 19\)

* **O** (bottom right-ish): \(17 \text{ \& } 51\)
* \(\text{HCF} = 17\)

* **N** (bottom-most left): \(25, 75 \text{ \& } 100\)
* \(\text{HCF} = 25\)

* **A** (bottom-most right): \(26 \text{ \& } 52\)
* \(\text{HCF} = 26\)

---

Step 2: Map HCF values to letters

Below is the summary of the letter-to-number code:

| HCF Value | Letter |
| :--- | :--- |
| **4** | S |
| **5** | K |
| **6** | H |
| **7** | A |
| **8** | O |
| **9** | E |
| **11** | I |
| **12** | E |
| **13** | N |
| **14** | F |
| **15** | T |
| **16** | A |
| **17** | O |
| **18** | H |
| **19** | K |
| **20** | A |
| **21** | F |
| **24** | M |
| **25** | N |
| **26** | A |
| **30** | R |
| **45** | D |

*(Note: The value \(2\) does not have a corresponding letter box on the sheet, but by looking at the word structure, it represents a space/separator between words).*

---

Step 3: Decode the puzzle

Now we substitute the letters into the two rows of boxes at the bottom of the page using their corresponding HCF values.

Row 1:

* **15** \( ightarrow\) **T**
* **6** \( ightarrow\) **H**
* **12** \( ightarrow\) **E**
* **2** \( ightarrow\) *(space)*
* **25** \( ightarrow\) **N**
* **7** \( ightarrow\) **A**
* **24** \( ightarrow\) **M**
* **9** \( ightarrow\) **E**
* **2** \( ightarrow\) *(space)*
* **8** \( ightarrow\) **O**
* **14** \( ightarrow\) **F**
* **2** \( ightarrow\) *(space)*
* **16** \( ightarrow\) **A**
* **2** \( ightarrow\) *(space)*
* **5** \( ightarrow\) **K**
* **11** \( ightarrow\) **I**
* **13** \( ightarrow\) **N**
* **45** \( ightarrow\) **D**

**Row 1 Decoded:** `THE NAME OF A KIND`

---

Row 2:

* **17** \( ightarrow\) **O**
* **21** \( ightarrow\) **F**
* **2** \( ightarrow\) *(space)*
* **20** \( ightarrow\) **A**
* **2** \( ightarrow\) *(space)*
* **4** \( ightarrow\) **S**
* **18** \( ightarrow\) **H**
* **26** \( ightarrow\) **A**
* **30** \( ightarrow\) **R**
* **19** \( ightarrow\) **K**

**Row 2 Decoded:** `OF A SHARK`

---

Answer:

THE NAME OF A KIND OF A SHARK