Can a cube end with exactly two zeroes (00)? Explain.
No, a cube cannot end with exactly two zeroes.
Reasoning: If a number ends with n zeros, its cube ends with 3n zeros.
Step 1: A number ending with n zeros can be written as k × 10ⁿ
Step 2: Its cube is (k × 10ⁿ)³ = k³ × 10³ⁿ
Step 3: The cube has 3n zeros at the end
Examples: 10³ = 1000 (3 zeros) 100³ = 1000000 (6 zeros)
Conclusion: Since 3n is always a multiple of 3, a cube can only end with 0, 3, 6, 9, ... zeros. ∴ A cube can never end with exactly 2 zeros.
Explanation
The chapter explains that when a number ending with zeros is cubed, the number of trailing zeros triples. This is because (k × 10ⁿ)³ = k³ × 10³ⁿ, giving 3n zeros at the end.
Solution Steps
Step 1: A number ending with n zeros can be written as k × 10ⁿ
Step 2: Its cube is (k × 10ⁿ)³ = k³ × 10³ⁿ
Step 3: The cube has 3n zeros at the end
Step 4: Since 3n is always a multiple of 3, it cannot equal 2