Question

i. Cube of any odd number is even.
ii. A perfect cube does not end with zeros.
iii. If a square of a number ends with 5, then its cube ends with 25.
iv. There is no perfect cube that ends with 8.
v. The cube of two digit numbers may be a three digit number.
vi. The cube of a two digit number may have seven or more digits.
vii. The cube of a single digit number may be a single digit number.

Answer 1

Hint: In order to state about whether the given statement is true or false, first we will consider an example with that and we will proceed further. We have to go through all the options one by one.

Complete step-by-step answer:
i. False
Example
$
  {3^3} = 27 \to {\text{ Odd}} \\
  {7^3} = 343 \to {\text{ Odd}} \\ $
The Cube of odd numbers is always odd.

ii. True
Example
\[
  {10^3} = 1000 \to 3{\text{ Zeroes}} \\
  {20^3} = 8000 \to 3{\text{ Zeroes}} \\
  {1300^3} = 2197000000 \to 6{\text{ Zeroes}} \\
 \]
A perfect cube does not end with 2 zeroes.

iii. False
Example
Since
$
  {15^2} = 275 \to {\text{Ends with 5}} \\
  {\text{1}}{{\text{5}}^3} = 3375 \to {\text{Doesn't end with 25}} \\ $
The given statement is false.

iv. False
Example
$
  {2^3} = 8 \to {\text{Ends with 8}} \\
  {12^3} = 1728 \to {\text{Ends with 8}} \\$
So, there are perfect cubes like 8 and 1728 that end with 8.

v. False
Example
The minimum two digits number is 10
And ${10^3} = 1000 \to 4{\text{ Digit number}}$
The maximum two digits number is 99
And ${99^3} = 970299 \to 6{\text{ Digit number}}$
So, the cube of a two digit number cannot be a 3 digit number.

vi. False
The minimum two digits number is 10
And ${10^3} = 1000 \to 4{\text{ Digit number}}$
The maximum two digits number is 99
And ${99^3} = 970299 \to 6{\text{ Digit number}}$
So, the cube of two digit numbers cannot have seven or more digits.

vii. True
Since
${1^3} = 1$
${2^3} = 8 $
The given statement is true

Note: In order to solve these types of questions read the statement carefully and try to find the example which contradicts the given statement because there are many statements which may satisfy the given statement but there are only few examples which contradict the given statements. Also you must have good knowledge of numbers and know how to calculate the cube and square of a number.