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1. To calculate reminder on dividing the number by 27 and 37

ANS:

Let me explain this rule by taking examples
consider number 34568276, we have to calculate the reminder on diving this number by 27 and 37 respectively.
make triplets as written below starting from units place
34.........568..........276
now sum of all triplets = 34+568+276 = 878
divide it by 27 we get reminder as 14
divide it by 37 we get reminder as 27

Example. 
other examples for the clarification of the rule
let the number is 2387850765
triplets are 2...387...850...765
sum of the triplets = 2+387+850+765 = 2004
on revising the steps we get
2......004
sum = 6
divide it by 27 we get reminder as 6
divide it by 37 we get reminder as 6

2. To calculate reminder on dividing the number by 7, 11, 13

ANS:

Let me explain this rule by taking examples
consider number 34568276, we have to calculate the reminder on diving this number by 7, 11, 13 respectively.
make triplets as written below starting from units place
34.........568..........276
now alternate sum = 34+276 = 310 and 568
and difference of these sums = 568-310 = 258
divide it by 7 we get reminder as 6
divide it by 11 we get reminder as 5
divide it by 13 we get reminder as 11 

Example. 
other examples:-
consider the number 4523895099854
triplet pairs are 4...523...895...099...854
alternate sums are 4+895+854=1753 and 523+099=622
difference = 1131
revise the same tripling process
1......131
so difference = 131-1 = 130
divide it by 7 we get reminder as 4
divide it by 11 we get reminder as 9
divide it by 13 we get reminder as 0

3. To calculate reminder on dividing the number by 3

ANS:
Method:- first calculate the digit sum , then divide it by 3, the reminder in this case will be the required reminder

example:- 1342568
let the number is as written above
its digit sum = 29 = 11 = 2
so reminder will be 2 

Example. 
Take some others
34259677858
digit sum of the number is 64 = 10 = 1
reminder is 1

similarly let the number is 54670329845
then digit sum = 53 = 8
when we divide 8 by 3 we get reminder as 2 so answer will be 2

4. Square of numbers near to 100

ANS:
Let me explain this rule by taking examples
96^2 :-
First calculate 100-96, it is 4
so 96^2 = (96-4)----4^2 = 9216
similarly
106^2 :-
First calculate 106-100, it is 6
so 106^2 = (106+6)----6^2 = 11236 

Example. 
An other case arises
110^2 = (110+10)----100 = (120+1)----00 = 12100
similarly
89^2 = (89-11)----121 = (78+1)----21 = 7921

5. Square of any 2 digit number

ANS:

Let me explain this trick by taking examples
67^2 = [6^2][7^2]+20*6*7 = 3649+840 = 4489
similarly
25^2 = [2^2][5^2]+20*2*5 = 425+200 = 625
Take one more example
97^2 = [9^2][7^2]+20*9*7 = 8149+1260 = 9409
Here [] is not an operation, it is only a separation between initial 2 and last 2 digits 

Example. 
Here an extra case arises
Consider the following examples for that
91^2 = [9^2][1^2]+20*9*1 = 8101+180 = 8281

6. Multiplication of 2 two-digit numbers where the first digit of both the numbers are same and the last digit of the two numbers sum to 10 (You can’t use this rule for other numbers)

ANS:
Let me explain this rule by taking examples
To calculate 56×54: 
Multiply 5 by 5+1. So, 5*6 = 30. Write down 30.
Multiply together the last digits: 6*4 = 24. Write down 24.
The product of 56 and 54 is thus 3024. 

Example. 
Understand the rule by 1 more example
78*72 = [7*(7+1)][8*2] = 5616

7. Multiplication of two numbers that differ by 6 (This trick only works if you have memorized or can quickly calculate the squares of numbers.

ANS:
If the two numbers differ by 6 then their product is the square of their average minus 9.
Let me explain this rule by taking examples
10*16 = 13^2 - 9 = 160
22*28 = 25^2 - 9 = 616 

Example. 
Understand the rule by 1 more example
997*1003 = 1000^2 - 9 = 999991

8. Multiplication of 125 with any number (You can't use this rule for other numbers)

ANS:

Let me explain this rule by taking examples
1. 93*125 = 93000/8 = 11625. 
2. 137*125 = 137000/8 = 17125. 

Example. 
Understand the rule by 1 more example
3786*125 = 3786000/8 = 473250.

9. Square of a 2 digit number ending with 5 (You can't use this rule for other numbers)

ANS:

Let me explain this rule by taking examples
35^2 = 3*(3+1) ---25 = 1225 

Example. 
Other example
95^2 = 9*(9+1) ---25 = 9025