**Quantitative Aptitude Notes on Simplification**

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**Mathmatical Simplification**the

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**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)

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.

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:

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

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)

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