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## Arihant Maths Book For Competitive Exams PDF Free Download

### Arihant Maths

#### Multiplication

When a! is multiplied by ‘£>’, then ‘a’ is added ‘b’ times or ‘b’ is added ‘a’ times. It is denoted by ‘x’.

Let us see the following operation on Multiplication If a =2 and b = 4, then 2×4=8 or (2+ 2+ 2+ 2) =8 Here, ‘a! is added ‘b’ times, or in other words, 2 is added 4 times.

Similarly, 4×2=8 or (4+ 4) =8 In this case, ‘b’ is added ‘a! times, or in other words, 4 is added 2 times.

#### Divisibility Tests

Divisibility by 2 When the last digit of a number is either 0 or even, then the number is divisible by

2. For example 12, 86, 472, 520, 1000 etc., are divisible by 2. **Divisibility** by 3 When the sum of the digits of a number is divisible by 3, then the number is divisible by

3. For example (i) 1233 1 + 2 + 3 + 3 = 9, which is divisible by 3, so 1233 must be divisible by 3. (ii) 156 1 + 5 + 6 = 12, which is divisible by 3, so 156 must be divisible by

3. Divisibility by 4 When the number made by the last two digits of a number is divisible by 4, then that particular number is divisible by 4.

Apart from this, the number having two or more zeroes at the end is also divisible by 4. For example (i) 6428 is divisible by 4 as the number made by its last two digits i.e., 28 is divisible by 4.

(ii) The numbers 4300, 153000, 9530000, etc., are divisible by 4 as they have two or more zeroes at the end.

Divisibility by 5 Numbers having 0 or 5 at the end is divisible by 5.

For example, 45, 4350, 135, 14850, etc., are divisible by 5 as they have 0 or 5 at the end.

Divisibility by 6 When a number is divisible by both 3 and 2, then that particular number is divisible by 6 also.

For example, 18, 36, 720, 1440, etc., are divisible by 6 as they are divisible by both 3 and 2.

Divisibility by 7 A number is divisible by 7 when the difference between twice the digit at one’s place and the number formed by other digits is either zero or a multiple of 7.

For example, 658 is divisible by 7 because 65 – 2 X 8 = 65 – 16 = 49. As 49 is divisible by 7, the number 658 is also divisible by 7.

Divisibility by 8 When the number made by the last three digits of a number is divisible by 8, then the number is also divisible by 8.

Apart from this, if the last three or more digits of a number are zeroes, then the number is divisible by 8.

For example (i) 2256 As 256 (the last three digits of 2256) is divisible by 8, 2256 is also divisible by 8.

(ii) 4362000 As 4362000 has three zeroes at the end.

Therefore it will definitely be divisible by 8.

Divisibility by 9 When the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9.

For example (i) 936819 9+3 + 6 + 8 + 1 + 9= 36 which is divisible by Therefore, 936819 is also divisible by 9.

(ii) 4356 4 + 3 + 5 + 6 = 18 which is divisible by 9. Therefore, 4356 is also divisible by 9.

Divisibility by II When a number ends with zero, then it is divisible by 10.

For example, 20, 40, 150, 123450, 478970, etc., are divisible by 10 as these all end with zero.

Divisibility by 1′ When the sums of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.

For example (i) 2865423 Let us see the Sum of digits at odd places (A) = 2 + 6+4 + 3 = 15 Sum of digits at even places (B) = 8 + 5 + 2 = 15 =>A = B Hence, 2865423 is divisible by 11.

(ii) 217382 Let us see

Sum of digits at odd places (A) = 2 + 7 + 8 = 17

Sum of digits at even places (B) = 1 + 3 + 2 = 6

A- B = 17-6 = 11 Clearly, 217382 is divisible by 11.

Divisibility by 12 A number which is divisible by both 4 and 3 is also divisible by 12.

For example, 2244 is divisible by both 3 and 4. **Therefore**, it is divisible by 12 also.

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