Question 12

We will examine each statement about cube numbers and determine if it is true or false, providing clear reasoning and examples.

Step 1 — Cube of an odd number

An odd number is a whole number that cannot be divided exactly by 2. Let us check the cubes of some odd numbers.

$1^3 = 1 \times 1 \times 1$

$= 1$

$3^3 = 3 \times 3 \times 3$

$= 27$

$5^3 = 5 \times 5 \times 5$

$= 125$

$\boxed{\text{The cube of an odd number is always odd.}}$

When we multiply odd numbers together, the result is always an odd number. So, the statement is False.

Step 2 — Perfect cubes ending with 8

A perfect cube is a number obtained by multiplying an integer by itself three times. Let us look at the last digit of cubes of numbers.

$2^3 = 2 \times 2 \times 2$

$= 8$

$12^3 = 12 \times 12 \times 12$

$= 1728$

$\boxed{\text{Perfect cubes can end with 8.}}$

The number $2^3$ ends with 8. The number $12^3$ also ends with 8. So, the statement is False.

Step 3 — Cube of a 2-digit number (3 digits)

A 2-digit number is any whole number from 10 to 99. Let us find the cube of the smallest 2-digit number.

$10^3 = 10 \times 10 \times 10$

$= 1000$

$\boxed{\text{The smallest 2-digit number's cube is 1000.}}$

The number 1000 has 4 digits. Any 2-digit number larger than 10 will have a cube even larger than 1000. So, the cube of a 2-digit number will always have at least 4 digits. The statement is False.

Step 4 — Cube of a 2-digit number (7 or more digits)

Let us find the cube of the largest 2-digit number, which is 99. We know that $100^3$ is $1,000,000$, which has 7 digits.

$99^3 = (100 - 1)^3$

$= 100^3 - 3 \times 100^2 \times 1 + 3 \times 100 \times 1^2 - 1^3$

$= 1,000,000 - 3 \times 10,000 + 300 - 1$

$= 1,000,000 - 30,000 + 300 - 1$

$= 970,299$

$\boxed{\text{The largest 2-digit number's cube is 970,299.}}$

The number 970,299 has 6 digits. Since the cube of the largest 2-digit number has 6 digits, no cube of a 2-digit number can have seven or more digits. The statement is False.

Step 5 — Factors of cube numbers

Factors are numbers that divide another number exactly. Let us find the factors for some cube numbers.

Consider the cube number $8 = 2^3$. The factors of 8 are 1, 2, 4, and 8. There are 4 factors. This is an even number.

Consider the cube number $27 = 3^3$. The factors of 27 are 1, 3, 9, and 27. There are 4 factors. This is an even number.

$\boxed{\text{Cube numbers can have an even number of factors.}}$

A number has an odd number of factors only if it is a perfect square. Since not all cube numbers are perfect squares (like 8 and 27), not all cube numbers have an odd number of factors. The statement is False.

Answer

(i) False. The cube of an odd number is always odd. For example, $3^3 = 27$, which is odd. (ii) False. Perfect cubes can end with 8. For example, $2^3 = 8$ and $12^3 = 1728$. (iii) False. The smallest 2-digit number is 10, and its cube is $10^3 = 1000$, which has 4 digits. (iv) False. The largest 2-digit number is 99, and its cube is $99^3 = 970299$, which has 6 digits. (v) False. Cube numbers can have an even number of factors. For example, the cube number 8 has 4 factors (1, 2, 4, 8), which is an even number.