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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 co-prime.

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 co-prime 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)