To find factors of numbers the following rules are to be remembered. .

(a) A number to be divisible by 2 has to have its last digit an even number (128, 246)

(b) A number is divisible by 3 if the sum of its digits is 3 or its multiple. (123 = 1 + 2 + 3 = 6 hence divisible by 3, similarly 342, 789 etc,.) .

(c) A number is divisible by 4 if its last two digits are divisible by 4. (178564, 236784 etc.,) .

(d) A number is divisible by 5 if its last digit is 5 or zero. (15, 40, etc) .

(e) A number is divisible by 6 if it is divisible both by 2 and 3. (18, 42, 96 etc.) .

(f) A number is divisible by 8, if its last 3 digits are divisible by 8. (3816, 14328, 18864 etc.,) .

(g) A number is divisible by 9, if the sum of its digits equals 9 or is a multiple of 9. (18 = 8 + 1 = 9, hence divisible by 9. Similarly, 729, 981, etc.) .

(h) A number is divisible by 11, if the sum of the alternate digits starting from the units digit (U) is equal to the sum of the alternate digits starting from the tens digit (T) or (U - T), is a multiple of 11. (for 132, sum of alternate digits is 1 + 2 = 3 and 3 which are equal, hence it is a multiple of 11 (Examples: 1331, 2541) .

(i) A number is divisible by 19 if the sum of the number of tens and twice the units digit is divisible by 19. .

eg. 38 3 + (2 x 8) = 19, hence 38 is a multiple of 19. .

1292

&⇒ In 1292, No. of
tens is 129. .

Therefore, 129 + (2 x 2) = 133 ; 133

&⇒ 13 + 2 x 3 = 19
hence, 1292 is divisible by 19. .

(1) If
n is even, n(n^{2} + 20) is divisible by 48. .

(2) If
n is a prime, greater than 3, n^{2} - 1 is divisible by 24. .

(3) The difference between any number and its square is even. .

(4) n(n + 1)(n + 5) is divisible by 6. .

(5) any prime number can be written in the form of (6k+1) or (6k-1), where k = 1, 2, 3, 4, 5…………..

Involution is the general name of multiplying the expression by itself so as to find first, second, third ... powers. For a number, the square, cube etc. are the 2nd, 3rd, etc. powers of the number - i.e., the number multiplied by itself to certain power. .

It is worth noting that square of every
expression is positive. eg.:- (a^{2})^{3} = a^{2} . a^{2}
. a^{2} = a^{6} .

The following algebraic formulae should be learnt by heart and are very useful (there will be problems in the S.S.C. exam involving simplification using such formulae): .

(a + b)^{2} = a^{2}
+ 2ab + b^{2}

(a - b)^{2} = a^{2}
- 2ab + b^{2}

(a + b + c)^{2} = a^{2}
+ b^{2} + c^{2} +2ab + 2bc + 2ca

(a + b)^{3} = a^{3}
+ b^{3} + 3ab(a + b)

(a - b)^{3} =
a^{3} - b^{3} - 3ab(a - b)

a^{2} - b^{2} =
(a + b)(a - b)

a^{3} + b^{3} =
(a + b)(a^{2} - ab + b^{2})

a^{3} – b^{3 }=
(a - b)(a^{2} + ab + b^{2})

a^{3} + b^{3} + c^{3}
- 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} - ab -
bc - ca)

These formulae can be used effectively in finding out powers of some numbers. .

112^{2} = (100 + 12)^{2} =
100^{2} + (2 x 100 x 12) + 12^{2}

89^{2} = (100 - 11)^{ 2} =
7921

17 x 23 = (20 - 3)(20 + 3) = 20^{2 }-
3^{2}

17^{2} = (17 + 3)(17 - 3) + 3^{2 }=
20 x 14 + 9 = 289

39^{2 }= (39 + 1)(39 - 1) + 1^{2 }=
40 x 38 + 1^{2}

= 1520 + 1 = 1521

The root of any number or expression is that quantity which when multiplied by itself the requisite number of times produces the given expression. The operation of finding the root is evolution. .

(a) Any even root of a positive quantity may be negative or positive.

(b) No negative quantity can have an even root. .

(c) Every odd root will have the same sign as the quantity itself. .

A decimal in which a digit or a set of digits is repeated continually is called a Recurring decimal. Recurring decimals are written in a shortened form, the digits which are repeated being marked by dots placed over the first and the last of them, thus .

8/3 = 2.666 = 2.6; .

1/7 = 0.142857142857142857... = 0.142857 .

21/22 = 0.9545454……….. = 0.954 .

Such a decimal as 0.142857, in which all the digits recur, is called a pure circulator, and such a decimal as 0.954 in which some do not recur is called a mixed circulator. The digit, or set of digits, which is repeated is called the period of the decimal. .

In the decimal equivalent to 8/3, the period is 6 ; in 1/7 it is 142857 and in 21/22 it is 54. .

The General Rile will be considered later, but if we consider a few examples, we shall be able to see what the rule is