# Division and Multiplication

By 2: If the last digit of the given number is even or zero.

By 3: If the sum of the digits of the number is divisible by 3.

By 4: If the last two digits of the number are divisible by 4.

By 5: If the last digit of the number is either zero or 5.

By 6: If the number is divisible by 2 as well as 3.

By 7: A number is divisible by 7, if the number of tens added to five times the number of units is divisible by 7. e.g. if you check 259, number of tens = 25 and 5 times units digit 5 × 9 = 45 now 25 + 45 = 70. As 70 is divisible by 7 so this number is divisible by 7.

By 8: If the last three digits of the number are divisible by 8.

By 9: If the sum of the digits is divisible by 9.

By 11: If the difference between the sum of the digits at odd places and the digits at even places in a number is either zero or divisible by 11.

By 12: Check for divisibility of both 3 and 4.

By 13: If the number of tens added to four times the number of units is divisible by 13. Then the number is divisible by 13. e.g. 4394 no. of tens = 439, number of units = 4 × 4 = 16 ⇒ 439 + 16 = 455, which is divisible by 13, so the number is divisible by 13.

By 17: A number is divisible by 17, if the number of tens added to twelve times the number of units is divisible by 17.

By 19: A number is divisible by 19, if the number of tens added to twice the number of units is divisible by 19.

By 25: When the number formed by the last two digits of the number is divisible by 25. [475 is divisible by 25 but 465 is not].

By 125: When the number formed by the last three digits of the number is divisible by 125. [999625 is divisible by 125 but 999525 is not].

# Short cuts for multiplying

**Short cuts for multiplying**: Large multiplication should be avoided. Instead, look for shortcuts to do the sums.

To multiply by 99, 999, 9999...: Place as many zeroes after the number and subtract the number.

To multiply by 5n: Put n zeroes to the right of the number and divide it by 2n.

To multiply by using formulae: Try to convert the number into an algebraic formula.

(i) (a+b)2 =a2 +b2 +2ab; (ii) (a–b)2 =a2 +b2 +2ab; (iii) (a+b)2 –(a–b)2 =4ab;

(iv) (a+b)2 +(a–b)2 =2(a2 +b2); (v) (a2 –b2)=(a+b)(a–b)

(vi) (a+b)3 =a3 +b3 +3ab(a+b) (vii) (a–b)3 =a3 –b3 –3ab(a–b)

(viii) (a3 –b3)=(a–b)(a2 +ab+b2).

# Sum of natural numbers

Σn = n(n+1)/2

Where Σn = Sum of first n natural numbers.

For example Σ5 = (5 × 6)/2 = 15

Σ10 = (10×11)/2 = 55

**Illustration 8:**

Multiply 3718 by 9999.

As given above, we write 3718(10000 – 1)

= 37180000 – 3718 = 371476282.

**Illustration 9:**

Multiply 43986 by 625.

43986 × 625 = 43986 × 54 = 43986 ́ (10/2)4 = 43986 × 10000/24 = 439860000/16 = 27491250.
**Note: **We hope the student realizes the importance of learning tables and doing quick multiplication and division. One should practice well in this otherwise important time will be lost both in Maths as well as in the DI section.