Friday, 10 June 2016

Number System - Quantitative Aptitude

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Number System (Quantitative Aptitude) for Bank PO, Clerk, SSC CHSL, CGL and other competitive exams| Face Value and Place Value and Various types of numbers

Indian numeral system consist of ten digits namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In words, zero, one, two, three, four, five, six, seven, eight, nine, ten.

Numeral: Number is represented by group of digits called numeral.

Face Value and Place Value:

Face Value: It is the value of the digit itself. For example, 7 is always 7 whether it at 578 or 5887888. The face value of digit not change with the place it appears in numeral.

Place Value: It is the place value of the digit multiplied by the place value at which it appears in the numeral. For example:
♦ In numeral 85269
◘ The place value of 9 is (9 × 1) = 9
◘ The place value of 6 is (6 × 10) = 60
◘ The place value of 2 is (2 × 100) = 200
◘ The place value of 5 is (5 × 1000) = 5000
◘ The place value of 8 is (8 × 10000) = 80000

Various Types of Numbers:

Natural Numbers: Counting numbers are called natural numbers. Example: 1, 2, 3, 4, ... etc. are all natural numbers. Smallest natural number is 1 and largest natural number is ∞.

Whole Numbers: All counting numbers together with zero are called whole numbers. Example: 0, 1, 2, 3, 4, ... are whole number.
◘ All natural number are whole number.

Integers: All counting numbers, 0 and -ve of counting numbers are called integers. Example: -∞..., -4, -3, -2, -1, 0, 1, 2, 3, 4 ... ∞ are integers.
◘ Set of Positive integers: [1, 2, 3, 4, ....]
◘ Set of Negative integers: [-1, -2, -3, -4, ...]
◘ Set of all Non-Negative integers: [0, 1, 2, 3, 4, ....]

Real Numbers: Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.

Rational Numbers: Any number which can be expressed in the form of p/q where p and q both are integers and q 0 are called rational numbers. Example: 3/7, -2/8, -5 etc.
◘ There exists infinite rational numbers between any two rational numbers.

Irrational Numbers: Non-recurring and Non-terminating decimals are called irrational numbers. Example: √3, √5, etc.

Even Numbers: Numbers which are exactly divisible by 2 are called even numbers. Example: -4, -2, 0, 2, 4 etc.
◘ Sum of first n even numbers = n(n + 1)

Odd Numbers:  Numbers which are not exactly divisible by 2 are called odd numbers. Example: -5, -3, 1, 3, 5 etc.
◘ Sum of first n odd numbers = n2

Prime numbers: Numbers which have exactly 2 factors, 1 and number itself is called Prime Numbers. Example: 2, 3, 5, 7, 11, etc.
◘ 1 is neither prime nor composite.
◘ 2 is only even prime number.
◘ There are 15 prime numbers between 1 to 50 and 10 prime numbers between 50 to 100.

Composite numbers: Numbers which have more than 2 factors. Example: 4, 6, 8, 10, etc.

Relative Prime Numbers: Two numbers are said to be relatively prime if they do not have any common factor other than 1.Example: (3,5), (7,20), (11,15), etc.

Twin Prime Numbers: Two prime numbers which differ by 2 are called twin prime numbers. Example: (3.5), (5,7), (11,13) etc.

Co-Prime Numbers: Two natural numbers a and b are said to be co-prime if their HCF is 1. Example: (2,3), (4,5), (7,9) etc.

Divisibility Rules

• Divisibility rules let you test if one number is divisible by another, without doing calculations.

• Divisibility by 2: A number is divisible by 2 if its digit at ones place (unit digit) is 0, 2, 4, 6 or 8. Example: 5896, 8572, 4856, etc.

• Divisibility by 3: A number is divisible by 2 when sum of its digits is divisible by 3. 
Example: 5496 = 5 + 4 + 9 + 6 = 24; 24 is divisible by 3 therefore, 5496 is also divisible by 3.

• Divisibility by 4: A number is divisible by 4 if the sum of its last two digits is divisible by 4. Example: 58975 = 7 + 5 = 12; 12 is divisible by 4 therefore, 5496 is also divisible by 4. (In 58975, 7 and 5 are last digits).

• Divisibility by 5: A number is divisible by 5 if its digit at ones place (unit digit) is 0 or 5.
Example: 8565, 25840, 50, etc.

 Divisibility by 6: A number is divisible by 6 if it is divisible both by 2 and 3.
Example: 18, 24, 48, etc.

• Divisibility by 7: A number is divisible by 7 if its digit at ones place (unit digit)  or last digit is multiplied by 2 and subtracted from the rest of the number and result is either 0 or divisible by 7.
Example: 672 = 2 × 2 = 4, 67 - 4 = 63, 63 ÷ 7 = 9; Here, result 63 is divisible by 7 therefore 672 is also divisible by 7. 

• Divisibility by 8: A number is divisible by 8 if the number formed by the last 3 digits of the number is divisible by 8.
Example: 589160, 698552, 58256, etc. whose last three digits is divisible by 8. 160 ÷ 8 = 20.

• Divisibility by 9: A number is divisible by 9 if sum if all the digits of a number is divisible by 9.
Example: 585 = 5 + 8 + 5 = 18; 18 is divisible by 9 therefore, 585 is also divisible by 9.

• Divisibility by 10: A number is divisible by 10 if its digit at ones place (unit digit) is 0.
Example: 580, 5690, 63210, etc.

• Divisibility by 11: A number is divisible by 11 if the the difference between the sum of its digits in odd places and in even places is either 0 or divisible by 11.
Example: 2728 = 2-7+2-8 = -11. Here, -11 is divisible by 11 therefore, 2728 is also divisible by 11.

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