Question

What is the GCF of 16 and 128?

Answer 1

Hint: To find the GCF of 16 and 128, first we will find the factors of both 16 and 128 separately by using the prime factorisation method. Then, after that we can take out the common factors from both the numbers and this will give us the GCF.

Complete step by step answer:
Given numbers = 16 and 128
We have to find the GCF of these numbers. The full form of GCF is the greatest common factor. GCF can be defined as the product of prime factors which are common to the prime factorization of both numbers. For calculating the GCF of two numbers, we have to use the prime factorization method. In this method the number is written as the multiplication of its prime factors.
Prime factorization of 16 is as below,
$\begin{align}
  & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
= \[2\times 2\times 2\times 2\]
Prime factorization of 128 is as below,
\[\begin{align}
  & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}\]
= \[2\times 2\times 2\times 2\times 2\times 2\times 2\]
Now we will see the common factors in the prime factorization of both numbers.
Common factors = \[2\times 2\times 2\times 2\]
Common factors = \[16\]
According to the definition of GCF, it is the product of these common factors,
So, the GCF of \[16\] and \[128\] is 16.

Therefore, the correct answer is 16.

Note: GCF can also be found using another method known as long division method. In this method, the higher number is divided by a smaller number and in the next step the remainder is made the divisor and divisor is made as dividend and again division is performed. These steps are continued till we get the remainder as zero and the last divisor will be the GCF of both numbers. Some students have confusion between the terms HCF and GCF. The full form of HCF is the highest common factor and the full form of GCF is the greatest common factor. These both terms are actually the same. The meaning of both highest and greatest is the multiplication of all common prime factors.