State true or false. Explain your reasoning. (i) The cube of any odd number is even. (ii) There is no perfect cube that ends with 8. (iii) The cube of a 2-digit number may be a 3-digit number. (iv) The cube of a 2-digit number may have seven or more digits. (v) Cube numbers have an odd number of factors.
False. For example, 10 is the smallest 2-digit number, and 10³ = 1000 has exactly 4 digits.
Explanation
(i) Multiplying odd numbers always gives an odd result. (ii) Cubes can end with any digit from 0 to 9. (iii) The smallest 2-digit number is 10, and 10³ = 1000 has 4 digits. (iv) 99³ = 970299 has 6 digits, which is the maximum for any 2-digit cube. (v) False. The statement is false. The only numbers that have an odd number of factors are perfect squares. Most cube numbers have an even number of factors, unless the cube number happens to also be a perfect square (like , which is both and )