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ideal-classes-blog · 7 years

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Factors & Multiples

Finding HCF by Division Method Prime and Composite Numbers

An exact divisor of a number is called its factor.

Ex: 1, 2, 3 and 6 are factors of number 6.

The number 1 is a factor of every number. Every number is a factor of itself. The factors of a number are either less than or equal to the number itself. All numbers have a finite number of factors. The product of two numbers is called a multiple of each of the two numbers being multiplied. A number is a multiple of all its factors. Every number is a multiple of 1 and of itself. There are infinite multiples of a number. If the sum of the factors of a number is two times the number, then the number is called a perfect number. Numbers that have only two factors in the form of 1 and the number itself are called prime numbers.

Numbers that have more than two factors are called composite numbers. The number 1 is neither a prime number nor a composite number. All numbers with 0, 2, 4, 6 or 8 in the unit?s or one?s place are multiples of 2, and are called even numbers. All numbers with 1, 3, 5, 7 or 9 in the unit?s or one?s place are called odd numbers.

The number 2 is the smallest prime number, and also the only prime number that is even. All prime numbers, except 2, are odd numbers. The sum of any two prime numbers, except with 2, is an even number.

Divisibility of Numbers

Factor:An exact divisor of a number is called its factor.

Multiple: The product of two numbers is called a multiple of each of the two numbers being multiplied.

Prime numbers: Numbers with two factors, 1 and itself, are called prime numbers.

Tests of divisibility:

There are certain tests of divisibility that can help us to decide whether a given number is divisible by another number.

Divisibility of numbers by 2:

A number that has 0, 2, 4, 6 or 8 in its ones place is divisible by 2.

Divisibility of numbers by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility of numbers by 4

A number is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.

Divisibility of numbers by 5

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

Divisibility of numbers by 6:

A number is divisible by 6 if that number is divisible by both 2 and 3.

Divisibility of numbers by 8:

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility of numbers by 9:

A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility of numbers by 10:

A number that has 0 in its ones place is divisible by 10.

Divisibility of numbers by 11:

If the difference between the sum of the digits at the odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.

Co?prime numbers:

If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

General rules of divisibility for all numbers:

Prime Factorisation, HCF and LCM

Writing a number as a product of its prime factors is called the prime factorisation of the number. Eg: (i) 18=2 x 3 x 3 (ii) 40=2 x 2 x 2 x 5 The greatest of the common factors of the given numbers is called their highest common factor (HCF). It is also known as the greatest common divisor. Eg: Prime factorisation of 16 = 2 x 2 x 2 x 2 Prime factorisation of 40 = 2 x 2 x 2 x 5 HCF of 16 and 40 = 2 x 2 x 2 = 8 The smallest common multiple of the given numbers is called their Least Common Multiple (LCM). Eg: The LCM of given numbers using their prime factorisation: Prime factorisation of 4 = 2 x 2 Prime factorisation of 6 = 2 x 3 LCM of 4 and 6 = 2 x 2 x 3 =12

To find the LCM of the given numbers using the division method:

Write the given numbers in a row.

Divide the numbers by the smallest prime number that divides one or more of the given numbers.

Write the number that is not divisible, in the second row.

Write the new dividends in the second row.

Divide the new dividends by another smallest prime number.

Continue dividing till the dividends are all prime numbers or 1.Stop the process when all the new dividends are prime numbers or 1.

Finding HCF by Division Method

Consider the numbers 16 and 24. Factors:16 = 16 × 1 = 8 × 2 = 4 × 4Factors of 16: 16, 8, 4, 2 and 1.24 = 24 × 1 = 12 × 2 = 8 × 3 = 6 × 4Factors of 24: 24, 12, 8, 6, 4, 3, 2 and 1Common factors of 16 and 24 = 8, 4, 2 and 1Largest common factor = 8 ∴ HCF of 16 and 24 = 8The HCF of two or more numbers is a number which is:• A common factor of all the given numbers• The largest among the common factors.HCF is also known as Greatest Common Divisor, that is, GCD.To find the HCF of two or more numbers:• Determine all the factors of each number, and• Identify the largest among the common factors.Methods of finding HCF:• Continued Division method• Common Division methodContinued Division method:i. Divide the larger number by the smaller number.ii. If the remainder is 0, the smaller number, that is the divisor, is the HCF.ii. If the remainder is not 0, divide the divisor by the remainder, which is the new divisor.iv. If the new remainder is 0, the last divisor is the HCF.v. Otherwise repeat the process of dividing the divisor by the remainder till the remainder becomes zero. The last divisor is the HCF.e.g.The numbers 48 and 30.H.C.F of 48 and 30 is 6HCF for three or more numbers:Consider the numbers 135, 150 and 225.i. First, take 135 and 150 and to find HCF of these two numbers.HCF of 135 and 150 = 15ii. Next, take the HCF of the first two numbers obtained, 15 and the remaining number 225.HCF of 15 and 225 = 15HCF of 135, 150 and 225 = 15Consider the numbers 16, 24, 36 and 42.iii. Take any two numbers and find their HCF, say 16 and 24.iv. Take the HCF of the first two numbers obtained and a number from the remaining numbers, and find their HCF.v. Repeat the process till all the given numbers have been considered.The last HCF is the HCF of all the given numbers.NOTE: The HCF of the given numbers will be the same, irrespective of the order in which the numbers are taken for finding HCF.Common Division method:i. Find a prime factor that is common to all the given numbers.ii. Divide all the given numbers with the common prime factor.iii. Identify a common prime factor of the quotients.iv. Divide all the quotients with the common prime factor.v. Repeat this process until the quotients have no common prime factor.The product of the common prime factors is the HCF of all the given numbers.e.g.Consider the numbers 171, 189 and 243.HCF of 171, 189 and 243 = 3 x 3= 9Co-prime NumbersWhen the HCF of two natural numbers is 1, the numbers are called co-prime numbers.Consider the numbers 33 and 49.HCF of 49 and 33 = 1 49 and 33 are co-prime numbers.A few other examples of co-prime numbers arei. 4 and 9.ii. 5 and 12.

Relation Between LCM and HCF

The LCM of two or more numbers is the smallest natural number that is a multiple of each of the given numbers. The HCF of two or more numbers is the greatest number that divides each one of them exactly. Relation between the HCF and LCM of any two given numbers: Product of LCM and HCF of two given numbers is equal to the product of the numbers.

If p and q are two numbers, p × q = LCM of p and q × HCF of p and q. Formulas obtained by rearranging the equation: LCM of p and q = p × q HCF of p and q HCF of p and q = p × q LCM of p and q

HCF of fractions: The HCF of fractions is defined as (the HCF of the numerators) divided by (the LCM of the denominators). HCF of fractions = HCF of numerators LCM of denominators LCM of fractions: LCM of fractions is defined as (the LCM of the numerators) divided by (the HCF of the denominators). LCM of fractions = LCM of numerators HCF of denominators

#Divisibility of Numbers #Prime Factorisation HCF and LCM #Finding HCF by Division Method #Relation Between LCM and HCF #Prime and Composite Numbers #Factors And Multiples #Class 8 Mathematics #ICSE - Class 8 Maths

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maffsguru · 3 years

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Divisibility and Prime Factorisation | Year 8 Maths | MaffsGuru.com

Have you ever wondered what the divisibility rules are? Want to know how to find the HCF and LCM of numbers? Do you know that HCF stands for Highest Common Factor and LCM stands for Lowest Common Multiple? Want to know how to do Prime Factorisation using Factor Trees? Then this video will answer all those questions you may - or may not! - have. Lots of worked examples are provided with downloadable notes (at www.maffsguru.com).

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brightengineersacademysposts · 3 years

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