Question

Fill in the blanks:
$48 = 6 \times 8$so 6 and 8, are …………. of 48.

Answer 1

Hint: 48 is a composite number which can be represented as a product of prime factors other than 1 and 48, and that is how it is given that \[48 = 6 \times 8\]. We need to prove that 6 and 8 are factors of 48.

Complete step-by-step answer:
In the question, we have been given a composite number 48 as a product of 6 and 8, and asked to establish the relationship of 6 and 8 with 48.
Now, composite numbers are those numbers which have prime factors other than 1 and the number itself. So, according to question 48 has been expressed a product of composite factors- 6 and 8.
To find out the relationship between 6, 8 and 48, let us first perform the prime factorization of 48, which will be:
$
  2\left| \!{\underline {\,
  {48} \,}} \right. \\
  2\left| \!{\underline {\,
  {24} \,}} \right. \\
  2\left| \!{\underline {\,
  {12} \,}} \right. \\
  2\left| \!{\underline {\,
  6 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
  1\left| \!{\underline {\,
  1 \,}} \right. \\
$
Which can be written as:
$
  48 = 2 \times 2 \times 2 \times 2 \times 3 \times 1 \\
   \Rightarrow 48 = 16 \times 3 \\
$
The factors can be further arranged and 48 can be written as:
$
  48 = 8 \times 6 \\
  Or,48 = 24 \times 2 \\
  Or,48 = 12 \times 4 \\
  Or,48 = 48 \times 1 \\
$
Now, 48 , clearly on prime factorization has factors other than 1 and 48 itself, hence it's a composite number. With prime factors of 48 as 2 and 3, we can express 48 as more than one possible set of composite factors. From the above representation, we can see that one of the representation of 48 with composite factors is :
$48 = 6 \times 8$ which can be also written as $48 = 8 \times 6$ , such that the factors remain the same, just the order of factors is different.
The above statement means that 48 can be expressed as a product of its factors 6 and 8,
Hence, 6 and 8 are factors of 48 since 6 and 8 on multiplication will give 48 as a product.
Therefore 6 and 8 are factors of 48.
Thus, we will fill the blank with the word factors, such that the statement will become:
$48 = 6 \times 8$, so 6 and 8, are …factors… of 48.

Note: It should be kept in mind that prime number and prime factors are two different things. While prime number is one which has 1 and the number itself as the factors, for example, 2, 3 ,5 etc are such numbers which when factorized will have no other factor other than 1 and the number itself; prime factors, when factorized will have only 1 and themselves as the factors, that is why they are called prime factors. For example, if we consider the number 12, so $12 = 2 \times 2 \times 3 \times 1$ , that is, 12 can be expressed as a product of two prime factors 2 and 3 with 2 raised to the power of 2. So, when 12 is factorized it is expressed in terms of its prime factors$12 = 2 \times 2 \times 3 \times 1$. And, these prime factors when multiplied will make the same composite number 12.