Bank PO exam -Maths Tricks

 NUMBER SYSTEM

 A. TYPES OF NUMBERS

1. Natural Numbers : Counting numbers 1, 2, 3, 4, 5,..... are called natural numbers.

2. Whole Numbers : All counting numbers together with zero form the set of whole numbers. Thus,

(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.

3. Integers : All natural numbers, 0 and negatives of counting numbers i.e., 
{…, - 3 , - 2 , - 1 , 0, 1, 2, 3,…..} together form the set of integers.

(i) Positive Integers : {1, 2, 3, 4, …..} is the set of all positive integers.
(ii) Negative Integers : {- 1, - 2, - 3,…..} is the set of all negative integers. 
(iii) Non-Positive and Non-Negative Integers : 0 is neither positive nor negative.

So, {0, 1, 2, 3,….} represents the set of non-negative integers, while {0, - 1 , - 2 , - 3 , …..} represents the set of non-positive integers.

4. Even Numbers : A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, etc.

5. Odd Numbers : A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

6. Prime Numbers : A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself. 
Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. 

Prime numbers Greater than 100 : Let be a given number greater than 100. To find out whether it is prime or not, we use the following method : 
Find a whole number nearly greater than the square root of p. Let k > square root of p. Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.

e.g,,We have to find whether 191 is a prime number or not. Now, 14 > square root of 191. Prime numbers less than 14 are 2, 3, 5, 7, 11, 13. 
191 is not divisible by any of them. So, 191 is a prime number.

7. Composite Numbers : Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12.

Note : 
(i) 1 is neither prime nor composite. 
(ii) 2 is the only even number which is prime. 
(iii) There are 25 prime numbers between 1 and 100.

8. Co-primes : Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,

B. MULTIPLICATION BY SHORT CUT METHODS

1. Multiplication By Distributive Law : 
(i) a * (b + c) = a * b + a * c (ii) a * (b-c) = a * b - a * c.

Ex. 
(i)567958 x 99999 = 567958 x (100000 - 1) = 567958 x 100000 - 567958 x 1 = (56795800000 - 567958) = 56795232042. 
(ii)978 x 184 + 978 x 816 = 978 x (184 + 816) = 978 x 1000 = 978000.

2. Multiplication of a Number By 5ⁿ :

 Put n zeros to the right of the multiplicand and divide the number so formed by 2 ⁿ 

Ex. 975436 x 625 = 975436 x 5⁴= 9754360000 = 609647600

C. BASIC FORMULAE

(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 + c)² = a² + b² + c² + 2 (ab + bc + ca) 
(vii) (a³ + b³) = (a +b) (a² - ab + b²) 
(viii) (a³ - b³) = (a - b) (a² + ab + b²) 
(ix) (a³ + b³ + c³ -3abc) = (a + b + c) (a² + b² + c² - ab - bc - ca) 
(x) If a + b + c = 0, then a³ + b³ + c³ = 3abc.

D.DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM

If we divide a given number by another number, then : 
Dividend = (Divisor x Quotient) + Remainder

(i) (xⁿ - aⁿ ) is divisible by (x - a) for all values of n. 
(ii) (xⁿ - aⁿ) is divisible by (x + a) for all even values of n. 
(iii) (xⁿ + aⁿ) is divisible by (x + a) for all odd values of n.

E. PROGRESSION - A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.

1. Arithmetic Progression (A.P.) : If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P.

An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),... The nth term of this A.P. is given by T_n=a (n - 1) d. 
The sum of n terms of this A.P.

S_n = n/2 [2a + (n - 1) d] = n/2 (first term + last term).

SOME IMPORTANT RESULTS :

(i) (1 + 2 + 3 +…. + n) =n(n+1)/2 
(ii) (l² + 2² + 3² + ... + n²) = n (n+1)(2n+1)/6 
(iii) (1³ + 2³ + 3³ + ... + n³) =n²(n+1)²

2. Geometrical Progression (G.P.) : A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression. The constant ratio is called the common ratio of the G.P. A G.P. with first term a and common ratio r is : a, ar, ar²,

In this G.P. T_n = ar^n-1

Sum of the n terms, S_n = a(1-rⁿ) / (1-r)