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1) Number divisible by 2
Units digit – 0, 2, 4, 6, 8
Ex: 42, 66, 98, 1124
2) Number divisible by 3
Sum of digits is divisible by 3
Ex: 267 (2 + 6 + 7) = 15
15 is divisible by 3
3) Number divisible by 4
Number formed by the last two digits is divisible by 4
EX: 832
The last two digits is divisible by 4, hence 832 is divisible by 4
4) Number divisible by 5
Units digit is either zero or five
Ex: 50, 20, 55, 65, etc
5) Number divisible by 6
The number is divisible by both 2 and 3
EX: 168
Last digit = 8  (8 is divisible by 2)
Sum of digits = (1 + 6 + 8) = 15  (divisible by 3)
Hence, 168 is divisible by 6
6) Number divisible by 11
If the difference between the sums of the digits at even places and the sum of digits at odd places is either 0 or divisible by 11.
Ex: 4527039
Digits on even places: 4 + 2 + 0 + 9 =15
Digits on odd places: 5 + 7 + 3 = 15
Difference between odd and even = 0
Therefore, number is divisible by 11
7) Number divisible by 12
The number is divisible by both 4 and 3
Ex: 1932
Last two digits divisible by 4
Sum of digits = (1 + 9 + 3 + 2) = 15  (Divisible by 3)
Hence, the number 1932 is divisible by 12
Basic Formulae: (Must Remember)
1) (a  b)2 = (a2 + b2  2ab)
2) (a + b)2 = (a2 + b2 + 2ab)
3) (a + b) (a – b) = (a2 – b2 )
4) (a3 + b3) = (a + b) (a2 – ab + b2)
5) (a3  b3) = (a  b) (a2 – ab + b2)
6) (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
7) (a3 + b3 + c3 – 3abc) = (a + b + c) (a2 + b2 + c2 – ab – bc – ac)
Quick Tips and Tricks:
1) If H.C.F of two numbers is 1, then the numbers are said to be coprime.
To find a number, say b is divisible by a, find two numbers m and n, such that m*n = a, where m and n are coprime numbers and if b is divisible by both m and n then it is divisible by a.
2) Sum of the first n odd numbers = n2
3) Sum of first n even numbers = n ( n + 1)
4) Even numbers divisible by 2 can be expressed as 2n, n is an integer other than zero.
5) Odd numbers which are not divisible by 2 can be expressed as 2(n + 1), n is an integer.
6) Dividend = [(Divisor × Quotient)] + Remainder
7) If Dividend = an + bn or an – bn
a) If n is even: an  bn is divisible by (a + b)
b) If n is odd: an + bn is divisible by (a + b)
c) an  bn is always divisible by (a – b)
8) To find the unit digit of number which is in the form ab. (Ex: 7105, 9125)
1) If b is not divisible by 4
Step 1: Divide b by 4, if it is not divisible then find the remainder of b when divided by 4.
Step 2: Units digit = ar, r is the remainder.
2) If b is multiple of 4
Units digit is 6: When even numbers 2, 4, 6, 8 are raised to multiple of 4.
Units digit is 1: When odd numbers 3, 7 and 9 are raised to multiple of 4.
If there are 6 terms in a series, then find the sum of geometric series 2, 6, 18, 54, 
a. 758
b. 728
c. 754
d. 738
Correct Option:(b)
Find the number of terms in geometric progression 3, 6, 12, 24,  , 384.
a. 10 b. 11 c. 9 d. 8
Correct Option: (d)
If 6 + 12 + 18 + 24 +  = 1800, then find the number of terms in the series.
a. 21 b. 22 c. 23 d. 24
Correct Option: (d)
1 + 2 + 3 +  50 = ?
a. 1275 b. 1350 c. 1575 d. 1455
Correct Option: (a)
Find which of the following number is divisible by 11?
a. 246542 b. 415624 c. 146532 d. 426513
Correct Option: (b)
Find the largest 4 digit number which is divisible by 88.
a. 8844 b. 9999 c. 9944 d. 9930
Correct Option: (c)
Find the solution of (935421 × 625) = ?
a. 584638125 b. 524896335 c. 542879412 d. 582365890
Correct Option: (a)
The remainder is 29, when a number is divided 56. If the same number is divided by 8, then what is the remainder?
a. 3 b. 4 c. 7 d. 5
Correct Option : (d)


























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