**Category : **6th Class

** Number System**

**Learning Objective **

- Number System
- HCF
- LCM

** **

**Number System**

A system of naming or representing numbers.

** **

**Number**

A number is a mathematical object which is used to count, label and measure,

**Example **

1, 5, 19, 325

** **

**Main Type**

Natural numbers/whole-numbers, integers, rational numbers, irrational numbers, real numbers.

**Natural Numbers**

Counting numbers 1, 2, 3, 4, 5, 6... are called natural numbers. These numbers are also referred to as the positive integers.

**Properties**

- The set of natural numbers, commonly denoted by N.
- 1 is the smallest natural number.

- No largest natural number can be found because the set of natural numbers is infinite,

- The successor of a natural number is 1 more than the number.
- The predecessor of a natural number is 1 less than the number.

** Whole Numbers**

The natural numbers along with zero form the collection of whole numbers.

**Example**

0, 1, 2, 3, 4, 5, 6 ....

**Properties**

- The set of whole numbers, commonly denoted by W.
- 0 is the smallest whole number.

- No largest whole number can be found because the set of whole numbers is infinite.

- The successor of a whole number is 1 more than the number
- The predecessor of a whole number is 1 less than the number.

** **

**Factors**

A factor of a number is an exact divisor of that number.

**Example 1 **

Factors of \[4=1,\,2,\,4\]

**Example 2**

Factors of \[15=1,\,3,\,5,\,15\]

** **

**Properties**

- 1 is the factor of every number.
- Every number is a factor of itself.

- The factors of a number are smaller or equal to the number.

- Numbers of factors of a given number are finite.
- Every factor of a number is an exact divisor of that number.
- A number for which sum of all of its factors is equal to twice of twice the number is called a perfect number.

- 1 is the only number which has exactly one factor, namely itself.

** **

** **

**Multiple**

A multiple of any natural number is a number formed by multiplying that number by any whole number or a multiple is the product of any quantity and an integer or a number is said to be multiple of any of its factors.

**Example 1 **

First three multiple of 4 are \[4\times 1= 4,4\times 2 = 8\] and \[4\times 3 =12\]

**Example 2**

First three multiple of 19 are \[19\times 1=19,19\times 2=38\] and \[19\times 3=57\]

**Properties**

- 0 is the multiple of everything.
- Every number is a multiple of itself,

- Every multiple is greater than or equal to that number

- The number of multiples of a given number is infinite.
- A number is a multiple of each of its factors,

**Prime Numbers**

The numbers which have exactly two factors, 1 and the number itself, are called prime numbers.

**Example **

2, 3, 5, 7 etc. are example of prime numbers.

** **

**Properties**

- 2 is the lowest prime number
- 2 is the only even prime number

- Zero is not considered as a prime number.

- Every prime number except 2 is odd,

**Co-primes:** Two natural numbers which have only the common factor 1 are called co-primes.

**Example **

Some examples of pairs of co "primes are \[\left( 2,3 \right),\left( 5,7 \right),\left( 8,13 \right),\left( 17,26 \right),\left( 29,33 \right).\]

**Twin-primes:** Prime numbers that differ by 2 are called twin-primes.

**Example**

Some examples of pairs of twin-primes are \[\left( 3,5 \right),\left( 5,7 \right),\left( 11,13 \right),\left( 17,19 \right)\left( 29,31 \right).\]

** **

**Prime triplet:** A prime triplet is a set of three prime numbers of the form \[\left( P,P+2+P+6 \right)\] *or\[~\left( p,p+4,P+6 \right).\]*

**Example **

Some examples of prime triplets are \[\left( 5,7,11 \right),\left( 7,11,13 \right),\left( 11,13,17 \right),\left( 13,17,19 \right),\left( 17,19,23 \right),\left( 37,41,43 \right).\]

**Composite Number **

Numbers having more than two factors are called composite numbers.

**Example **

4, 6, 8, 9 are example of composite numbers.

** **

**Properties:**

- 4 is the smallest composite number
- 1 is neither a prime number nor the composite number

- All the even numbers except 2 are the composite numbers.

**Test for divisibility of numbers**

- If a number has 0 in the ones place then it is divisible by 10.

**Example:** \[20,40,50,100,200,260\] all are divisible by 10

- A number which has either 0 or 5 in its ones place is divisible by 5.

**Example:** 15, 20, 25, 30 are divisible by 5.

- A number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its ones place.

**Example:** 6, 24, 20, 34,102 are divisible by 2.

- If sum of the digits of a number is a multiple of 3 then the number Is divisible by 3.

**Example:** 9, 96,102, 201, 216 are divisible by 3.

- If a number is divisible by 2 and 3 both then the number is divisible by 6 also.

**Example:** 12, 24, 60, 36 are divisible by both 2 and 3 and also by 6.

- A number with three or more digits is divisible by 4 if the number formed by its last two digits (i.e., ones and tens) is divisible by 4.

**Example:** 324, 428, 520, 672 are divisible by 4

- A number with three or more digits is divisible by 8 if the number formed by its last three digits (i.e., ones, tens and hundreds) is divisible by 8.

**Example:** 1000, 1728, 13824, 8184 are divisible by 8

- If sum of the digits of a number is a multiple of 9 then the number is divisible by 9.

**Example:** 981, 729,108, 21456 are divisible by 9

- Find the difference between the sum of the digits at odd places (from the right) and sum of the digits at even places (from the right) of the number. If the difference is either 0 or multiple of 11 then the number is divisible by 11.

**Example:** 132, 1452, 7172, 2277 are divisible by 11

**HCF (highest common factor)**

The HCF of two or more numbers is the greatest number which divides each number exactly.

**Example **

HCF of 9 and 12 is 3 because 3 is the highest common factor among all the common factors of 9 and 12.

**Properties**

- HCF of two or more numbers is not greater than any of the given numbers.
- The HCF of two co-primes is 1.

- The HCF of a group of number is always a factor of their LCM.

- The product of HCF and LCM of two number is equal to product of the given numbers.

** **

** LCM (lowest common multiple)**

The LCM of two or more given numbers is the lowest (or smallest) of their common multiples.

**Example **

LCM of 18 and 24 is 72 because 72 is the smallest common multiple among all the common multiples of 18 and 24.

**Properties**

- The LCM of two or more given numbers is not less than any of the given numbers.
- The LCM of two co-primes is equal to their product.

- The LCM of a group of numbers is always a multiple of their HCF.

**Integers **

It includes the counting number \[\left( 1,1,~3,4.... \right)\], zero (0) and the negative of counting number \[\left( -1,-2,-3,-4... \right)\] So we can denote integers like this \[\left( ...-4,-3,-1,-1,0,1,2,3,4... \right).\] Denotation of integers on the number line has been shown below.

**Properties**

- Integers are numbers with no fractional part.
- The numbers \[+1,+2,+3...\] are positive integers and denoted by \[{{Z}^{+}}.\]

- The numbers \[-1,-2,-3...\] are negative integers and denoted by \[Z-.\]

- Zero (0) is neither a positive number nor a negative number.
- The negative of a negative integer is a positive integer.
- The negative of a positive integer is a negative integer.

**Example **

**Write all the integers between -3 and 4**

**Explanation:** The integers between -3 and 4 are \[-2,-1,0,1,2\] and 3.

**Fractions**

A fraction is a number which represents a part of a whole. The whole may be a single object or a group of objects. A fraction is written in the form a/b where a is called numerator of the fraction and b is called denominator of the fraction.

**Example **

\[\frac{5}{7},\frac{9}{8}\] and \[\frac{11}{13}\] are examples of fractions.

**Proper fraction:** A fraction whose numerator is smaller than its denominator is called a proper fraction.

**Example **

\[\frac{5}{7},\frac{5}{4}\] and \[\frac{54}{59}\] are examples of proper fractions.

**Improper fraction: **A fraction whose numerator is greater than its denominator is called an improper fraction.

**Example **

\[\frac{9}{8},\frac{15}{13}\] and \[\frac{21}{19}\] are examples of improper tractions.

**Mixed fraction:** A combination of a whole number and a proper fraction is called a mixed fraction.

**Example **

\[5\frac{1}{3},2\frac{1}{2}\] and \[3\frac{1}{2}\]are examples of mixed fractions.

**Equivalent fraction:** Fractions having same value are called equivalent tractions.

**Example **

\[\frac{4}{5},\frac{8}{10}\] and \[\frac{16}{20}\]are examples of equivalent fractions.

**Like fractions:** The fractions having same denominators are called like fractions.

**Example **

\[\frac{4}{5},\frac{8}{10}\] and \[\frac{6}{7}\]are examples of like fractions.

** **

**Unlike fractions: **The fractions having different denominators are called unlike fractions,

**Example **

\[\frac{2}{3},\frac{4}{5}\] and \[\frac{6}{7}\] are examples of unlike fractions.

**Decimals **

The numbers which use a decimal point followed by one or more digits are called decimal numbers.

**Example **

\[4.25,3.2,0.698\] are examples of decimal numbers.

** **

**Properties **

- Each decimal number has two parts: whole number part and decimal part,
- The number before the decimal point is called whole number part whereas the number after the decimal point is called the decimal part.

- Like decimals have an equal number of digits to the right of the decimal point

- Unlike decimals have an unequal number of digits to the right of the decimal point.

**Commonly Asked Questions **

** Which one of the following options is correct?**

(a) \[\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=1\] (b) \[\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=0-1\]

(c) \[\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=0\] (d) All of these

(e) None of these

** **

**Answer (c)**

**Explanation: \[\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=\frac{4}{3}-\frac{4}{3}=0\]**

** **

**Simplify: \[\mathbf{45+3\times 2}\,\mathbf{of}\,\mathbf{5-(16+4)-8\div 4}\]**

(a) 55 (b) 43

(c) 53 (d) Both (a) and (c)

(e) None of these

**Answer (c)**

**Explanation:** \[45+3\times 2\text{ }of\,5-\left( 16+4 \right)-8\div 4\]

\[=45+3\times 2\text{ }of\text{ }5-20-8\div 4\] [Bracket removed]

\[=45+3\times 2\text{ }of\text{ }5-20-2\] [Operation of division \[8\div 4=2\]]

\[=45+30-20-2\] [Operation of multiplication \[3\times 2\times 5=30\]]

\[=75-20-2\] [Operation of addition \[45+30=75\]]

\[=75-22=53\] [Operation of subtraction].

** **

**In the following picture, some parts of picture are shaded but some are not. What part of the picture is unshaded?**

(a) \[\frac{2}{5}\] (b) \[\frac{1}{5}\]

(c) \[\frac{3}{5}\] (d) All of these

(e) None of these

**Answer (b)**

**Explanation:** One part of the picture is not shaded but 4 parts are shaded.

** **

**Find the product of decimals 234.567 and 123.7.**

(a) 29016.9372 (b) 29015.9373

(c) 29016.1853 (d) 29015.9379

(e) All of these

**Answer: (d)**

**Explanation:** \[234.569\times 123.7=29015.9379.\]

** **

**Write the shortest form of the following: \[\mathbf{4000+500+8+0+}\frac{\mathbf{7}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{6}}{\mathbf{1000}}\mathbf{.}\]**

(a) 4508.706 (b) 4507.705

(c) 4509.707 (d) All of these

(e) None of these

**Answer (a)**

**Explanation:** \[4000+500+8+0+\frac{7}{10}+\frac{6}{1000}=4508.706.\]

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