Quantitative Aptitude Notes: Simplification

Quantitative Aptitude Notes on Simplification

BASIC FORMULAE

1. (a+b) ²=a²+b²+2ab 

2. (a−b) ²=a²+b²−2ab

3. (a +b) ²− (a−b) ²=4ab   

4. (a+b) ²+ (a−b) ²=2(a²+b²)

5. (a²–b²)= (a+b) (a−b)  

6. (a+b+c) ²=a²+b²+c²+2(ab+bc+ca)

7. (a³+b³) = (a+b) (a²−ab+b²)    

8. (a³–b³) = (a−b) (a²+ab+b²)

9. (a³+b³+c³−3abc)= (a+b+c) (a²+b²+c²−ab−bc−ca)

10. If a+b+c=0, then a³+b³+c³=3abc.

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, ex. 2, 4, 6, 8, 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.

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.

REMAINDER AND QUOTIENT:

"The remainder is r when p is divided by k" means p=kq+r the integer q is called the quotient.

EVEN ,ODD NUMBERS

A number n is even if the remainder is zero when n is divided by 2: n=2z+ 0 or n=2z.

A number n is odd if the remainder is one when n is divided by 2: n=2z+1.

even X even = even

odd X odd = odd

even X odd = even

even + even = even

odd + odd = even

even + odd = odd

Some important tricks

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
2. (1² + 2² + 3² + ..... + n²) = n ( n + 1 ) (2n + 1) / 6
3. (1³ + 2³ + 3³ + ..... + n³) = (n(n + 1)/ 2)²
4. Sum of first n odd numbers = n²
5. Sum of first n even numbers = n (n + 1)