Question 5

Step 1 · Prime Factorise Numbers in the Puzzle

Finding the prime factorisation for each number in the left figure and counting all prime factors (including repeats):

• 105: $105 = 3 \times 5 \times 7 \quad (\text{3 prime factors})$

• 20: $20 = 2 \times 2 \times 5 \quad (\text{3 prime factors})$

• 30: $30 = 2 \times 3 \times 5 \quad (\text{3 prime factors})$

• 28: $28 = 2 \times 2 \times 7 \quad (\text{3 prime factors})$

• 125: $125 = 5 \times 5 \times 5 \quad (\text{3 prime factors})$

• 18: $18 = 2 \times 3 \times 3 \quad (\text{3 prime factors})$

Step 2 · Prime Factorise Numbers in the Solution

Finding the prime factorisation for each number in the right figure:

• 8: $8 = 2 \times 2 \times 2 \quad (\text{3 prime factors})$

• 105: $105 = 3 \times 5 \times 7 \quad (\text{3 prime factors})$

• 70: $70 = 2 \times 5 \times 7 \quad (\text{3 prime factors})$

• 30: $30 = 2 \times 3 \times 5 \quad (\text{3 prime factors})$

• 70: $70 = 2 \times 5 \times 7 \quad (\text{3 prime factors})$

• 28: $28 = 2 \times 2 \times 7 \quad (\text{3 prime factors})$

Step 3 · Deduce the Rule

Every number in the yellow boxes has exactly 3 prime factors when counted with multiplicity (i.e., counting repeated prime factors as well).

Therefore, the rule to solve the puzzle is that each valid box must contain a number having exactly 3 prime factors.