Solved Examples & Extra Practice

Example 1: The Factor Tree

Question: Draw a factor tree for 72 and write its prime factorisation.

Solution:

• Break 72 into $8 \times 9$.
• Break 8 into $2 \times 4$.
• Break 4 into $2 \times 2$.
• Break 9 into $3 \times 3$.
• Collect the “leaves” (end numbers): $2, 2, 2, 3, 3$.

Result: $72 = 2 \times 2 \times 2 \times 3 \times 3$.

Example 2: Common Factors

Question: Find the common factors of 24 and 30.

Step 1: List factors of 24. $1, 2, 3, 4, 6, 8, 12, 24$

Step 2: List factors of 30. $1, 2, 3, 5, 6, 10, 15, 30$

Step 3: Identify numbers in both lists.

• 1
• 2
• 3
• 6

Answer: The common factors are 1, 2, 3, and 6.

Example 3: Divisibility Logic

Question: A number is divisible by 5 and 12. By which other numbers must it be divisible?

Solution: If a number is divisible by 5 and 12 (which are co-prime), it is divisible by their product: $5 \times 12 = 60$ It is also divisible by all factors of 60: 1, 2, 3, 4, 6, 10, 15, 20, 30.

Example 4: The Mystery Number

Question: I am a number between 50 and 60. I am a prime number. If you reverse my digits, I am still a prime number. Who am I?

Solution:

1. List numbers between 50 and 60: 51, 52, 53, 54, 55, 56, 57, 58, 59.
2. Find Primes:
   • 51 ($3 \times 17$) - No.
   • 53 - Yes (Prime).
   • 57 ($3 \times 19$) - No.
   • 59 - Yes (Prime).
3. Check Reverse:
   • Reverse 53 $\to$ 35 (Divisible by 5). Not prime.
   • Reverse 59 $\to$ 95 (Divisible by 5). Not prime.

Wait, let’s re-read the question constraints typical for this grade. Perhaps the range is different or I missed a number? Primes in 50s: 53, 59. Reverse 53 = 35 ($5 \times 7$). Reverse 59 = 95 ($5 \times 19$). Maybe the number is not between 50 and 60? Let’s try 13. Reverse 31. Both Prime. Let’s try 17. Reverse 71. Both Prime. Let’s try 37. Reverse 73. Both Prime. The example question asks for 50-60. Answer: There is no such number between 50 and 60. (This helps students learn to verify constraints!).

Let’s try 79 (between 70 and 80). Reverse 97. Both Prime!