An odd number is an integer which is not a multiple of two. 
For example 1,3,5,7...

Each option: 1 Mark
A number is divisible by 4 if the last two digits of the whole number are divisible 4.
(a) Since number formed by tens and units digit is 96, which is divisible by 4. Hence, 4096 is divisible by 4.
(b) Since number formed by tens and units digit is 84, which is divisible by 4. Hence, 21084 is divisible by 4.

Rule: 1 Mark
Solution: 1 Mark
A number is said to be divisible by 9 if the sum of its digits is divisible by 9.
The given number is 621.
Now, 6+2+1=9, which is divisible by 9.
$\therefore$ 621 is divisible by 9.

Steps: 2 Marks
Solution: 1 Mark
Every 4th student is wearing cap i.e. it is divisible by 4.

And every 5th is wearing a coat. i.e. it is divisible by 5.
So, if we want to find out which student is wearing both cap and coat, we need to find the LCM of 4 and 5.

LCM of 4,5 = 4 $\times$ 5 = 20. Since they are co-prime numbers, their LCM will be the product of these numbers. 

Hence, if a number is divisible by two co-prime numbers then it is divisible by their product also.

Steps: 2 Marks
Solution: 1 Mark
The lowest number which will be divisible by both of these numbers is the LCM of these numbers.
To find the LCM of the numbers first we will find out its prime factors
The  factors of 7: 1, 7

The factors of 12: 1, 2, 3, 4, 6, 12 

Since the only common factor is 1, the given two numbers are co-prime.
The LCM of these numbers is their product.
$\therefore$ The required number is 7$\times$12 = 84.

Concept: 1 Mark
Steps: 1 Mark
Answer: 1 Mark
The LCM of 3, 4 and 5 = 3 $\times$ 4 $\times$ 5 = 60
Therefore, the required number is 2 more than 60.
Hence, the required least number = 60 + 2 = 62.

Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark
We need to understand that if Rashi has to plant the same number of trees of the same type in a row, then the number of trees in each row should be a factor of 54 and 27.
Now, the greatest common factor will give the greatest number of trees Rashi can plant in each row.
The prime factorization of the numbers is given below.

54= 2 $\times$ 3 $\times$ 3 $\times$ 3

27= 3 $\times$ 3 $\times$ 3

So, the HCF will be 3 $\times$ 3 $\times$ 3 = 27

Therefore 27 trees can be planted in each row.
Total number of rows which will be planted = 54 $÷$ 27 + 27$÷$27 = 3

Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark
For the shortest height, we need to find the LCM of 12 and 18

The prime factorization of 12 and 18 are:

12 = 2 $\times$ 2 $\times$ 3;

18 = 2 $\times$ 3 $\times$ 3

In these prime factorizations, the maximum number of times the prime factor 2 occurs is two; this happens for 12.
Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18.
The LCM of the two numbers are the product of the prime factors counted the maximum number of times they occur in any of the numbers.
Thus, in this case LCM = 2 $\times$ 2 $\times$ 3 $\times$ 3 = 36.
Hence, the shortest height for which the two stacks will have the same height is 36 inches.

Definition: 1 Mark
Steps: 2 Marks
Answer: 1 Mark
A number for which sum of all its factors is equal to twice the number is called a perfect number.
We have to find out if 496 is a perfect number or not.
To find if a number is a perfect number or not first we need to find out all the factors of the number.
So,
1$\times$496=496
2$\times$248=496
4$\times$124=496
8$\times$62=496
16$\times$31=496
The various factors of 496 are 1,2,4,8,16,31,62,124,248,496.
Now sum of all the factors= 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992 = 2$\times$496
Hence, the sum of all the factors of the of 496 is twice the number. 
$\therefore$496 is a perfect number.

Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark
To check the divisibility of a number by 11, the rule is to find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.

3 + 9 = 12, A + 8

A + 8 - 12 = 0 
A - 4 = 0 or A - 4 = 11

A = 4 or A= 15, A = 15 is not possible.

So, the number is 3498.