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Class IV - maths: Factors and Multiples
One Word Answer Questions:
Q) Find the first 3 multiples of 7 ?
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Q) Find the first 5 multiples of 12 ?
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Q) Find the common factors of 12 and 14 ?
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Q) Is 78 is multiple of 9 ?
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Q) Write the all multiples of 4 which lie between 11 to 40 ?
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Q) Check the number 7645 is Divisible by 4 or not?
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Q) Find the common factor of 4 and 28 ?
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Q) The digit in the once place is 5 or 0 then the number is divisible by__________
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Q) Prime numbers have only____________factors.
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Q) 4758 is divisible by____.
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Q) ______is a factor of every number.
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Q) All multiples of 10 ends with a_______.
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Q) The smallest factor of a number is___?
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Short Answer Questions:
Q) Find the factors of 52 by division?
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Q) Find the factors of 64 by multiplication?
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Q) Find the Lcm of 56,8 ?
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Q) Find the Lcm of 9,12 ?
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Q) Write the first five multiples of 13?
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Q) What are the 4th,7th and 9th multiples of 14?
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Q) Use multiplication to find the factors of 24?
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Q) Find the prime factorization of 24 using factor tree method?
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Long Answer Questions:
Q) Find the prime factorisation of 72 using factor tree method?
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Q) Find the HCF and common factors of 25,50,125?
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Q) Find the HCF and LCM of 4,8 and 108?
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Q) Write down the all prime numbers in between 1 to 100 ?
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Q) Find out the CF,LCM and HCF of 9, 56 and 34 ?
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Q) Fill in the blanks 3,6,9,_____,________,18,_____,24.
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Q) Find the common factors of 14 and 18?
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Q) Find the first three common multiples of 6 and 12.What is their LCM?
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Q) Which of the following number is divisible by 11?
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Q) Find out the prime numbers between 1 to 9?
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Q) Find the factors of 16 by multiplication?
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Q) Find the Lcm and HCF of 126,14?
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Factors & multiples

    Definition
    Factor: "A factor is a whole number multiplied by another number to find the product".
    Multiple: "A multiple is a number which can be divided by another number without a remainder".

    Example:

    Factors and Multiples
    Finding Factors

    Factors of a number can be found by either using division or multiplication.

    Example 1: Find the factors of 32 by division.
    Solution:

    Factors and Multiples

    All the divisors and quotients are the factors of the number. When factors are repeated, no further division takes place.
    Thus, the factors of 32 in the ascending order are,
    1, 2, 4, 8, 16 and 32.

    Example 2: Find the factors of 18 by multiplication.
    Solution: We express 18 as a product of two factors(factor pair).
    To go systematically, we write the products in serial order.

    Factors and Multiples

    4 is not a factor of 18.

    So, the factors of 18 in ascending order = 1, 2, 3, 6, 9, and 18.

    Factors & multiples

    From the above examples, we can observe these facts about factors.

    1. 1 is a factor of every number
    2. The biggest factor of a number is the number itself
    3. 1 is the only number which has only one factor
    4. A factor of a number is smaller than or equal to a number
    5. A number has a limited number of factors.
    Facts about Multiples
    1. The first multiple of every number is the number itself
      Example: Multiples of 6 are:
      6 12 18 24......
      1st multiple 2nd multiple 3rd multiple 4th multiple.....
      Clearly, the 1st multiple of 6 is 6 itself
      Similarly, the 1st multiple of 8 is 8; 12 is 12; 80 is 80 etc.
      This also includes that every counting number is a multiple of itself.
    2. Multiples of a number have no last multiple as they can carry on and on.
      They are unlimited
      Example: Multiples of 30 are 30, 60, 90, 120.....
    3. Every number is a multiple of 1.
      Example: 1 × 1 = 1; 1 × 8 = 8; 1 × 450 = 450.
      But, 1 is the multiple of only the number 1.
      Example: 1 × 1 = 1

    Common Multiples and Least Common Multiples

    Common Multiples

    Consider the numbers 2 and 3.
    Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...
    Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30,...
    Observe the coloured numbers in the lists of multiples of 2 and 3
    Common multiples = 6, 12, 18.
    The common multiples are not confined to these numbers, the list of multiples continues.
    Let us find the common multiples of 2, 3 and 4
    Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...
    Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24....
    Multiples of 4 = 4, 8, 12, 16, 20, 24....
    Common multiples = 12, 24.....

    If we continue to find more multiples of these numbers, more common multiples will be found.

    Least Common Multiples

    Let us find the least common multiple (LCM) of the given numbers.
    Example: Look at the multiples of 4 and 6.
    Multiples of 4 = 4, 8, 12, 16, 20, 24....
    Multiples of 6 = 6, 12, 18, 24, 30....
    Common multiples = 12, 24....
    Least common multiple (LCM) = 12.

    To Check If The Bigger Number Is A Multiple Of The Smaller Number

    The bigger number must be divided exactly by the smaller number to be its multiple.
    Example 1: Is 54 a multiple 6?
    Solution: To find, we divide 54 by 6. We see that, 54 ÷ 6

    Factors and Multiples

    So, 54 is a multiple of 6.
    Example 2: Is 47 a multiple of 9?
    Solution: Let us divide 47 by 9

    Factors and Multiples

    Since, 47 is not divisible by 9, it is not a multiple of 9.

    Relation between Factors & multiples
    1. We know that 1 is the only number which has only 1 factor.
      1 × 1 = 1.
      So, 1 is the only factor and multiple of 1.
    2. Some numbers have only 2 factors
      Number 5 has two factors, 1 and 5(1 × 5 = 5)
      The product of these factors is their multiple.
    3. Numbers can have more than 2 factors.
      Let's look at the factors of 12.
      1 × 12 = 12; 2 × 6 = 12; 3 × 4 = 12;
      1 and 12; 2 and 6; 3 and 4 are factor pairs of 12.
      The product of the factor pairs is the multiple of the factors.
    4. Factors are less than or equal to the multiple of the number.
    5. The number itself is the smallest multiple and the greatest factor of itself.
Common Factors And Highest Common Factor

To find common factors, we find the factors of two, three or more numbers.
Example: Find the common factors of 16 and 18.

Factors of 16 Factors of 18
1×16=16 1×18=18
2×8=16 2×9=18
4×4=16 3×6=18
8×2=16 6×3=18

Factors of 16 = 1, 2,4,8, 16.....
Factors of 18 = 1, 2, 3, 6, 9, 18.....
Common factors of 16 and 18 = 1, 2.
Each of the common factors divides both 16 and 18.
Observe that the highest common factor (HCF) from the list of common factors of 16 and 18 is 2.

So, the HCF of two or three given numbers is the number which divides each of the numbers exactly.

Relationship of HCF and LCM

Take two numbers 2 and 18.
We know that, the smaller number can divide the bigger number exactly.
18 ÷ 2 = 9
If the smaller number divides the larger number without leaving a remainder, the smaller number is the HCF and the bigger number is the LCM of the two numbers.
Let us consider one more pair of numbers,

Example: 2 and 16
Solution: Divide the bigger number by smaller number and find the LCM and HCF.
16 ÷ 2 = 8. No remainder is left.
So, HCF = 2 and LCM = 16.

Tests Of Divisibility

The word divisibility relates to division without actually doing division.
Tests of divisibility help us to take a look at a number and find out if it can be divided by 2, 3, 5 and so on.
Tests of divisibility also help us to find out factors of a number.
Take a look at the following tests.

Factors and Multiples
Prime and Composite Number

Numbers can be expressed in many ways

  • Natural numbers - 1, 2, 3, 4, 5,6.....
  • Whole numbers - 0, 1, 2, 3, 4, 5.....
  • Even numbers - 2, 4, 6, 8, 10, 12....
  • Odd numbers - 1, 3, 5, 7, ....
  • Consecutive numbers - 1, 2, 3, 4, 5, 6 .....i.e.,.Which come one after the other.

    Now, let us take a look at prime and composite numbers.

    Prime Numbers:

    The numbers which have only two factors are called prime numbers.

    Composite Numbers:

    The numbers which have more than two factors, i.e. three and more factors are called composite numbers.

    Discovering Prime and Composite Number

    The process of factorization with the help of tests of divisibility helps us to discover the prime and composite numbers.
    Look at the factors of natural numbers from 1 to 10.

    Factors and Multiples

    From the above table, we observe that:

    1. 1 is the only number with 1 factor It is called a unique number or special number. It is neither prime nor composite.
    2. 2, 3, 5,7 have two factors. So, they are prime numbers between 1 and 10. Also, 2 is the smallest and the only even prime number.
    3. 4,6, 8, 9, 10 are composite numbers between 1 to 10. 4 is the smallest composite number.
    4. 2 and 3 are consecutive prime numbers.
    5. 3 and 5 or 5 and 7 are prime numbers with one composite number in between them. Such prime numbers are called twin prime numbers or twin primes.
    6. When you factorise two numbers and find only 1 as the common factor, such numbers are called co-prime numbers or co-primes.

    A few pair of co-primes is: 7 and 8; 5 and 9; 8 and 13; etc.

Prime Factorisation

Factorisation

Writing a number as a product of its factors is called 'factorisation'.

Prime Factorisation

A factorization in which every factor is a prime number is called the 'prime factorisation' of the number. Observe these examples:

Example 1: Find the prime factorisation of 36 using factor tree method?
Solution:

Factors and Multiples

The prime factorization of 36 = 3 × 3 × 2 × 2.

Example 2: Find the prime factorization of 48 using factor tree method.
Solution:

Factors and Multiples

The prime factorization of 48 = 2 × 2 × 2 × 2 × 3.

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