This GSEB Class 6 Maths Notes Chapter 3 Playing with Numbers covers all the important topics and concepts as mentioned in the chapter.
Playing with Numbers Class 6 GSEB Notes
Introduction:
We have studied about factors, multiples, prime and composite numbers in previous classes. We shall review these concepts and extend our study to include some new properties with suitable examples.
Factor:
A factor of a number ‘a’ is a number ‘b’ which completely divides a.
- 1 is a factor of every number.
- Every number is a factor of itself.
- Every number (other than 1) has atleast two factors, 1 and number itself.
- Every factor of a number is always less than or equal to the number.
- A number has always finite number of factors.
Multiple:
A multiple of a number ‘a ’ is a number obtained by multiplying ‘a’ by a natural number.
- Every number is a multiple of itself.
- Every multiple of a number is greater than or equal to the number.
- The smallest multiple of a natural number is the number itself.
- There are infinite multiples of a number. So the largest multiple can not be defined.
Perfect number:
If the sum of all the factors of a number is two times the number then the number is called a perfect number.
The factors of 6 are 1, 2, 3 and 6
Also, 1 + 2 + 3 + 6 = 12 = 2 × 6
i. e. sum of all factors of 6 = 2 × Number
So, 6 is the perfect number.
Even numbers:
All multiples of 2 are called even numbers.
Example: 2, 4, 6, 8, 10, are all even numbers.
Odd Numbers:
Numbers that are not multiples of 2 are called odd numbers.
Example: 1, 3, 5, 7, 9, 11, are all odd numbers.
Prime Numbers:
A natural number greater than 1, which has exactly two factors, namely 1 and itself, is called a Prime Number.
Example : 2, 3, 5, 7, 11, 13, 17, 19, are prime numbers.
Composite Numbers:
A natural number having at least one factor, besides 1 and itself, is called a composite number.
Example : 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, are Composite numbers.
Important facts:
- The number 1, has only one factor, namely 1, so it is neither Prime nor Composite.
- No natural number is both Prime and Composite.
- 2 is the only even prime number. All other even numbers are composite numbers.
- If two numbers do not have a common factor other than one, they are called Co-Prime.
Example : (3, 4), (2, 3) (5, 7) are Co-Prime numbers.
Common Factors and Multiples:
Common factor:
A whole number that divides exactly the two or more given numbers is called common factor.
For example : 7 is the common factor of 14, 49 and 84.
Common Multiple:
A whole number that is a multiple of each of a group of numbers is called common multiple.
For example: 100 is the common multiple of 5, 25 and 50.
Co-Prime Numbers:
Two numbers are said to be co-prime if they do not have a common factor other than 1.
For example : Common factors of 16 and 25 is 1. So these are called co-prime numbers.
- Two co-prime numbers need not to be both prime numbers.
- Two prime numbers are always co-prime.
Tests For Divisibility of Numbers:
Divisibility by 2:
A number is divisible by 2 if its unit digit is 0, 2, 4, 6, or 8.
Example : 540, 642, 754, 866 are all divisible by 2.
Dvisibility by 3:
A number is divisible by 3 if the sum of digits is exactly divisible by 3.
Example : 5463 is divisible by 3 because 5 + 4 + 6 + 3 = 18 which is divisible by 3.
Divisibility by 4:
A number is divisible by 4 if the last two (ten’s-units) digits of a number are zero or exactly divisible by 4.
Example: 500,7488 are divisible by. Here 88 + 4 = 22
Divisibility by 5:
A number is divisible by 5 if the unit’s digit is 0 or 5.
Example : 120, 125 are divisible by 5.
Divisibility by 6:
A number is divisible by 6 if it is divisible by 2 and 3.
Example : 594 is divisible by 6 because it is divisible by 2 and 3.
Divisibility by 8:
A number is divisible by 8 if the last 3 digits form a number divisible by 8.
Example: 56328 is divisible by 8 because 328 + 8 = 41
Divisibility by 9:
A number is divisible by 9 if sum of the digits is divisible by 9.
Example : 1872 is divisible by 9 because 1 + 8 + 7 + 2=18 which is divisible by 9.
Divisibility by 11:
A number is divisible by 11 if the sum of one set of alternate digits and the sum of other set of alternate digits differ by 0 or 11 or multiple of 11.
Example: Take the number 8050314052 Sum of digits at odd places
= 8 + 5 + 3 + 4 + 5 = 25 Sum of digits at even places
= 0 + 0 + 1 + 0 + 2 = 3 Difference of two sums = 25 – 3 = 22 which is divisible by 11. .’. given number is also divisible by 11.
Prime Factorisation:
Prime Factorisation is the process by which a composite number is rewritten as the product of prime factors.
Fundamental Theorem of Arithmetic.
Every Composite number can be factorised into prime factors in one and only one way apart from the order of the factors.
Prime factorization can be done by two methods :
- Factor Tree Method
- Division Method
1. Factor Tree Method:
In each step of the factor tree, we write the given composite number as the product of its smallest prime factors and another untill we get all the prime factors.
2. Division Method:
Let us find the prime factors of 180 using the division method.
- Step I. Divide the number by any prime number which will exactly divide it.
- Step II. Continue dividing the quotient by any prime number till we get the quotient itself as a prime number.
Highest Common Factor (HCF):
The highest common factor of two or more given numbers is the greatest among all their common factors.
Remember:
- If one number is the factor of the other, the smaller number is the HCF of the given numbers.
- HCF of two or more numbers can never be zero because 1 as a factor will be common to all numbers.
- HCF of two co-prime numbers is always 1.
- HCF is always smaller than or equal to the smallest of the given numbers.
There are two common methods to find H.C.F. of two or more numbers.
- Prime Factorisation Method
- Continued Division Method.
Here, we shall learn about these two methods.
1. Prime Factorisation Method:
To find H.C.F., we follow the following steps :
- Step 1: Make the prime factors of each of the given number.
- Step 2. Find the common prime factors of the given numbers.
- Step 3. The product of all common factors (of step 2) is the H.C.F. of given numbers.
2. Continued Division Method (Euclid’s Algorithm):
Euclid was a Greek mathematician. He derived an interesting method of find H.C.F. of two or more numbers. This method is known as Euclid’s algorithm or Long division method.
Euclid’s Algorithm (steps for finding H.C.F.)
- Step 1. From the given numbers, Identity the greater number.
- Step 2. Take the greater number as dividend . and the smallest number as divisor.
- Step 3. Find the quotient and remainder.
- Step 4. If the remainder is zero then the divisor is the required H.C.F.
- Step 5. If the remainder is non-zero then take the remainder as new divisor and the last divisor as the new dividend.
- Step 6. Repeat the steps till the remainder obtained is zero.
- Step 7. The last divisor for which the remainder is zero is the required H.C.F. Let us perform an activity based on this method.
Lowest Common Multiple (L.C.M.):
The lowest common multiple of two or more given numbers is the smallest among all their common multiples.
Remember:
If 1 is the only common factor of two given numbers, then the L.C.M. is the product of two numbers.
There are two methods to find L.C.M. of two or more numbers :
- Prime Factorisation Method
- Common Division Method
Now, we shall learn about these two methods :
1. Prime Factorisation Method. To find L.C.M., we follow the following steps :
- Step 1. Make the prime factors of each of the given number.
- Step 2. Find the product of all different prime factors with maximum number of times each factor appear.
- Step 3. The product of those factors is the required L.C.M.
2. Common Division Method. To find L.C.M. of two or more numbers, we follow the following steps :
- Step 1. Arrange the given numbers in a row separated by commas.
- Step 2. Obtain a number which divides exactly atleast two of the given numbers.
- Step 3. Write the quotients just below them which are divisible by the chosen number and carry forward the numbers which are not divisible by that number.
- Step 4. Repeat the process till no two of the given numbers divisible by the same number.
- Step 5. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.
Relation between H.C.F. and L.C.M.
- H.C.F. of given numbers is always a factor of L.C.M. or L.C.M. is a multiple of H.C.F.
- The product of H.C.F. and L.C.M. of two numbers is equal to product of both given numbers. If a and b are two numbers then axb = H.C.F. × L.C.M. Hence, Product of two numbers
= Product of their H.C.F. and L.C.M. Note : This result is true only for two numbers.