Question

Find the greatest common factor between \[24\]and\[112\]

Answer 1

Hint: We first need to find the prime factors of given numbers. Then we have to find the common factors in both. After that, we need to multiply the common factors to get the greatest common factor between the given two numbers.

Complete step by step answer:
The given numbers are \[24\]&\[112\]. We aim to find the greatest common factor between these two numbers.
We first need to find the prime factors of \[24\]&\[112\].
First, let us find the prime factorization of \[24\]
\[
  2\left| \!{\underline {\,
  {24} \,}} \right. \\
  2\left| \!{\underline {\,
  {12} \,}} \right. \\
  2\left| \!{\underline {\,
  6 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
 \]
Thus, the number \[24\]can be written as\[2 \times 2 \times 2 \times 3\]. That is \[24 = 2 \times 2 \times 2 \times 3\]
Now let us find the prime factors of \[112\]
\[
  2\left| \!{\underline {\,
  {112} \,}} \right. \\
  2\left| \!{\underline {\,
  {56} \,}} \right. \\
  2\left| \!{\underline {\,
  {28} \,}} \right. \\
  2\left| \!{\underline {\,
  {14} \,}} \right. \\
  7\left| \!{\underline {\,
  7 \,}} \right. \\
 \]
Thus, the number\[112\] can be written as\[2 \times 2 \times 2 \times 2 \times 7\]. That is \[112 = 2 \times 2 \times 2 \times 2 \times 7\]
Now let us compare the prime factors of both numbers to find the common factors.
On comparing \[24 = 2 \times 2 \times 2 \times 3\]& \[112 = 2 \times 2 \times 2 \times 2 \times 7\]we get that \[2 \times 2 \times 2\]as common factors. On multiplying this we get\[2 \times 2 \times 2 = 8\].
Thus, the greatest common factor of the given two numbers \[24\]&\[112\] is\[8\].

Note: The factors are nothing but the number when multiplied together gives another number, for instance, \[8\]can be written as \[2 \times 4\]then \[2\& 4\]are the factors of eight. A number is said to be a common factor when it is a factor of two or more numbers, for instance, the factors of \[12\]are \[1,2,3,4,6,12\]& the factors of \[30\]are \[1,2,3,5,6,10,15,30\]now we can see that \[1,2,3,6\]are the number common in both the factors thus, these are the common factors of\[12\] &\[30\]. We can also find the greatest common factor by finding the common factors that are mentioned above then the highest common factor will be the greatest common factor of those two numbers therefore the greatest common factor of \[12\] &\[30\] is\[6\].