Numbers & Algebra – Quantitative Aptitude Notes

Numerals

1 Lakh = 100,000

1 Crore = 10,000,000 = 100 Lakh

1 Million = 1,000,000 = 10 Lakh

1 Billion = 1,000,000,000 = 1000 Million = 100 Crore

1 Trillion = 1,000,000,000,000 = 1000 Billion

Face Value & Place Value

In the number ,

Face Value = 2,4,5,8 respectively.

Place Value = 2000, 400, 50, 8 respectively.

Types of Numbers

• Positive Numbers: -1,-2,-3,-4, …
• Negative Numbers: 1,2,3,4, ….
• Even Numbers: Numbers divisible by 2 – 0,2,4,6,8,…
• Odd Numbers: Numbers not divisible by 2 – 1,3,5,7,….
• Natural Numbers: Counting Numbers from 1,2,3,4, and so on. (0 excluded as it is not a counting number)
• Whole Numbers: Numbers from 0,1,2,3,4, and so on.
• Integers: All numbers from negative to 0 to positive …,-3,-2,-1,0,1,2,3,….
• Prime Numbers: A Number that is divisible by only 1 and itself. E.g. 2,3,5,7,11,13, etc.
• Composite Numbers: Numbers that are not prime, ie. numbers that are divisible by at least one number other than 1 and itself.

Tests of Divisibility

1. Divisibility by 2: Unit place should be an even number.
2. Divisibility by 3: The sum of all digits of a number should be divisible by 3.
3. Divisibility by 4: Last two digits should be divisible by 4.
4. Divisibility by 5: Units place should be 0 or 5.
5. Divisibility by 6: Number should be divisible by both 2 & 3.
6. Divisibility by 8: Last 3 digits should be divisible by 8.
7. Divisibility by 9: Sum of all digits should be divisible by 9.
8. Divisibility by 10: Units place should be 0.

Important Identities

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. a² – b² = (a + b)(a – b)
4. (a + b)³ = a³ + 3a²b + 3ab² + b³
5. (a – b)³ = a³ – 3a²b + 3ab² – b³
6. a³ – b³ = (a – b)(a² + ab + b²)