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Class V - maths: Factors and Multiples
One Word Answer Questions:
Q) What is a number which divides another number without leaving a remainder?
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Q) Which number is greater than 1 and has only two factors, 1 and the number itself?
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Q) Which number has more than two factors?
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Q) 4, 6, 8, 9, 10, etc., are which numbers?
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Q) 2, 3, 5, 7, 11, 13 etc., are which numbers?
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Q) Find the factors of 30?
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Q) Find the LCM of 20,5?
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Q) Find the lowest common multiple of 3 and 6?
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Q) A factor is a number which divides another number without leaving a?
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Q) A multiple is a number which can be divided by another number, without leaving a?
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Q) Write down the first ten multiples of 8?
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Q) Find last 5 multiples of 5?
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Short Answer Questions:
Q) Write the Properties of Multiples?
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Q) Pick out the prime numbers from the following 17, 8, 21, 24, 41, 18, 26?
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Q) Pick out the composite numbers from the following 47, 49, 51, 59, 61, 63, 65?
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Q) By using division method find the prime factorisation of the 120?
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Q) Find the HCF of 16, 20, 24?
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Q) What is Divisibility?
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Q) What is Factors?
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Q) What are the properties of factors?
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Q) Let us factories 168 by the division method?
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Q) Find the HCF of 18 and 24?
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Q) Find the LCM of numbers 2 and 7 through multiples?
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Long Answer Questions:
Q) Find the HCF, if the product of the two numbers is 720 and their LCM is 36?
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Q) Find the largest number which is a factor of 180 and 336?
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Q) What is the smallest number that is divisible by 20, 48 and 72?
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Q) The product of the HCF and LCM of two numbers is 1280. If one number is 32, find the other number?
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Q) Find the LCM of 12, 16, and 20 by the common division method?
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Q) What is Factors and give an example?
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Q) What is Divisibility and give an example?
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Q) What is Prime and Composite Numbers and give an example?
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Q) What is highest common factor and give an example?
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Q) What is lowest common factor and give an example?
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Q) What is multiples and give an example?
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Q) What is lowest common multiple and highest common factor?
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Q) Find the HCF, if the product of the two numbers is 810 and their LCM is 27?
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Factors and Multiples
Factors

"A factor is a number which divides another number without leaving a remainder".

Example: 16 ÷ 2 = 8 and 16 ÷ 8 = 2.
Both 2 and 8 are factors of 16.
We can find all the factors of a number in two ways:
  1. BY MULTIPLICATION
  2. Factors and Multiples
  3. BY DIVISION
Factors and Multiples

Here, 1, 2, 4, 8, and 16 are all factors of 16.

Rectangles can be used to find the factor pairs of a number.Let us draw all possible rectangles having 16 square units to find the factors of 16.

Factors and Multiples
PROPERTIES OF FACTORS
  • It is a factor of every number.
  • Every non-zero number is a factor of itself.
  • The smallest factor of a number is 1.
  • The largest factor of a number is the number itself.
  • Every factor of a non-zero number is less than or equal to the number.
Divisibility
Tests of Divisibility

Divisibility tests helps us to discover whether a number has a factor besides 1 and the number itself.

Let us learn the divisibility rule of 2, 3, 5,9, 10, 11.

DIVISIBILITY BY RULE EXAMPLES
2 The digit in the ones place should be 2, 4, 6, 8 or 0.
Th H T O
3 1 2 6
5 4 9 8
3 The sum of the digits of the number should be divisible by 3.
2457 = 2 + 4 + 5 + 7 = 18
73296 = 7 + 3 + 2 + 9 + 6 = 27
5 The digit in the ones place should be 5 or 0.
Th H T O
2 4 8 0
9 6 3 5
9 The sum of the digits of the number should be divisible by 9.
45279 = 4 + 5 + 2 + 7 + 9 = 27
135 = 1 + 3 + 5 = 9
10 The digit in the ones place should be 0.
Th H T O
8 3 4 0
2 9 6 0
11 The difference of the sum of the alternate digits of a number should be either 0 or a multiple of 11.
520971
2 + 9 + 1 = 12
5 + 0 + 7 = 12
12 - 12 = 0.
Prime and Composite Numbers
PRIME NUMBER:

A number which is greater than 1 and has only two factors, 1 and the number itself is called a prime number. Examples: 2, 3, 5, 7, 11, 13 etc., are prime numbers.

COMPOSITE NUMBER:

A number which has more than two factors is called a composite number.

Examples:

4, 6, 8, 9, 10, etc., are composite numbers.

PRIME NUMBERS BETWEEN 1 AND 100

Following steps are to be followed to find the prime numbers between 1 and 100.

To find out the prime and composite numbers, we use the method derived by the Ancient Greek Mathematician named Eratosthenes.

Write the numbers from 1 to 100.
  • Make a crown on 1 because it is a unique number.
  • Encircle 2 and cross all the numbers divisible by 2.
  • Encircle 3 and cross all the numbers divisible by 3.
  • Encircle 5 and cross all the numbers divisible by 5.
  • Encircle 7 and cross all the numbers divisible by 7.
  • Encircle 11 and cross all the numbers divisible by 11.
  • Continue this process till all the numbers are either crossed-out or encircled.

All the encircled numbers are prime numbers and the crossed-out numbers except 1 are composite. The prime numbers between 1 and 100 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,79,83, 89, & 97

This method is known as the SIEVE OF ERATOSTHENES.

Factors and Multiples
TWIN PRIMES:

Two prime numbers with a composite number in between are called twin primes. Example: 3 and 5, 11 and 13 are twin primes.

There are two methods of prime factorisation.

  1. Factor tree method
  2. Division method.
FACTOR TREE METHOD

In this method, we factorise a composite number till we get all prime factors.

Example: Let us factorise number 36 using the factor tree method.

Solution:
Factors and Multiples
Prime factorization of 36 = 2 × 2 × 3 × 3.
DIVISION METHOD

In this method, we start dividing the given number by the smallest prime number and continue division by prime numbers till we reach 1.

Example: Let us factorise 168 by the division method.

Solution: Here, the division is done starting with the smallest prime number and continuing till we reach the number 1.

Factors and Multiples
So, the prime factorisation of 168 = 2 × 2 × 2 × 3 ×7.
HIGHEST COMMON FACTOR (HCF)

Definition:

"The greatest number which divides two or more numbers exactly without leaving any remainder is called the highest common factor or HCF".

Example:Find the HCF of 18 and 24.

Solution:

Factors of 18 = 1, 2, 3, 6, 9 and 18.

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24.

We can notice that, 1, 2, 3, and 6 are the factors which are common in both the numbers 18 and 24. Out of these common factors, 6 is the greatest factor which is common to both 18 and 24.

So, we can say that 6 is the HCF of 18 and 24.

HCF OF 3 NUMBERS

To find the HCF of three numbers, we first find the separate factors of the 3 numbers. Then take out the common factors of which HCF is identified.

Example: Find the HCF of 8, 12, and 16.

Solution:
FACTORS OF 8 FACTORS OF 12 FACTORS OF 16
1 × 8 = 8 1 × 12 = 12 1 × 16 = 16
2 × 4= 8 2 × 6 = 12 2 × 8 = 16
3 × 4 = 12 4 × 4 = 16
  • Factors of 8 = 1, 2, 4 and 8.
  • Factors of 12 = 1, 2, 3, 4, 6 and 12.
  • Factors of 16 = 1, 2, 4, 8 and 16.
  • Common factors = 1, 2 and 4.

Highest common factor = 4.

ALTERNATIVE METHOD

HCF BY SHORT DIVISION METHOD:

Factors and Multiples

HCF = 2 × 2 = 4 (Multiply all common prime factors).

We can find HCF by two methods.

  1. Prime factorisation method
  2. Division method
Factors and Multiples
Factors and Multiples
MULTIPLES

"A multiple is a number which can be divided by another number, without leaving a remainder".

Example 1: Find the multiples of 6.

Solution: We know that,

6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24

Here, 6 have been multiplied by 1, 2, 3, and 4 consecutively to get the products as 6, 12, 18, and 24 respectively. So, we can say that the multiples of 6 = 6, 12, 18, 24 and so on.

Similarly,
  • Multiples of 2 = 2, 4, 6, 8
  • Multiples of 3 = 3, 6, 9, 12
  • Multiples of 4 = 4, 8, 12, 14

The multiples of a number are obtained by multiplying the number consecutively by each of the natural numbers.

Example 2: Write down the first ten multiples of 8.

Solution:The first ten multiples of 8 are: 8, 16, 24, 32, 40,48, 56, 64, 72, 80.

PROPERTIES OF MULTIPLES
  • Every number is a multiple of itself
  • Every number is a multiple of 1.
  • Every multiple of a number is greater than or equal to the number.
  • The smallest multiple of a number is the number itself.
  • The multiples of an even number are always even numbers.
  • The multiples of an odd number are alternatively odd and even numbers.
LOWEST COMMON MULTIPLE

"The smallest number that can be divided by the given numbers without leaving any remainder is called the lowest common multiple".

Example: Find the lowest common multiple of 3 and 4.

Solution:
Factors and Multiples

In the above example, out of the common multiples, 12 is the lowest.

So, we can say that the LCM = 12.

Lowest Common Multiple is also called the Least Common Multiple and is written as LCM.

LCM THROUGH MULTIPLES

Example: Find the LCM of numbers 2 and 7 through multiples.

Solution: We know that,

  • Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18.
  • Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56.
Factors and Multiples

The first common multiple of both the numbers = 14.

So, LCM of 2 and 7 = 14.

LCM OF THREE NUMBERS

Example: Find the LCM of three numbers4 , 6 and 8.

Solution: First write a few multiples of all the three numbers and then take out the common multiples and pick up their LCM.

Factors and Multiples

So, the LCM of 4, 6, and 8 = 24.

LCM BY PRIME FACTORISATION

In this method, we first list the prime factors of the numbers and then multiply the common factors and the remaining prime factors.

Example:Find the LCM of 24 and 36 using prime factorization method.

Solution:
Factors and Multiples

Multiplying the common factors 2, 2, and 3, and the remaining prime factors, we get: LCM = 2 ×2× 3 × 2 × 3 = 72.

LCM BY COMMON DIVISION METHOD

Example: Find the LCM of 20, 25 and 30 using the common division method.

Solution:

The following steps are followed to find the LCM using common division method.

  • Divide by the smallest prime number, which can divide atleast one of the numbers, and bring down the numbers that cannot be divided further.
  • Continue division by the smallest possible prime numbers till the last row contains prime numbers or co-prime numbers.
  • Multiply all the factors and the numbers in the last row to get the LCM.
Factors and Multiples

LCM of 20, 25 and 30 = 2 × 2 ×3 × 5 × 5 = 300.

RELATIONSHIP OF HCF AND LCM

The product of HCF and LCM of two natural numbers is equal to the product of the two numbers.

Example: Find the HCF and LCM of 60 and 72, and also find their relation.

Solution:
Factors and Multiples
Examples:
  1. Find the greatest number which divides 300 and 396 exactly.
  2. Solution: To find the greatest number, we have to find the HCF.

    Factors and Multiples

    So, last divisor = 12, i.e., HCF = 12.

    So, 12 is the greatest number that divides the given numbers exactly.

  3. Find the smallest number that is divisible by 18, 24, and 36.
  4. Solution:To find the smallest dividend, we have to find the LCM.

    Factors and Multiples
    LCM = 2 × 2 × 3 × 3 × 2 = 72.

    So, 72 is the smallest number that is exactly divisible by the given numbers.

  5. The HCF of two numbers is 16 and the LCM is 42. If one number is 24, find the other number.
  6. Solution: We know that,

    Product of two numbers = HCF × LCM

    So, 1st number × 2nd number = HCF × LCM

    1. 2nd number = HCF × LCM/ 1st number
    2. 2nd number = 16 × 42/24
    3. 2nd number =672/24
    4. 2nd number = 28.
    5. The other number = 28.
  7. Find the HCF, if the product of the two numbers is 810 and their LCM is 27.
  8. Solution: We know that:

    1. HCF × LCM = Product of two numbers
    2. HCF = Product of two numbers / LCM
    3. HCF = 810 / 27
    4. HCF = 30.
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