Question

Which of the following are the prime factors of 143.
A. $2\times 11\times 13$
B. $11\times 13$
C. $2\times 3\times 7$
D. None of these

Answer 1

Hint: To find the prime factors of 143, we will divide 143 with each prime number starting from 2. We will skip the prime numbers that result in a whole number when dividing 143 by it. The prime number that can exactly divide 143, that is, without giving a whole number, will be considered. That is, $\dfrac{143}{11}=13$ . Now, we will divide the result of this by the prime numbers starting from 2 and do the same procedure. That is $\dfrac{13}{2}=6\dfrac{1}{2}$ . We will skip the prime numbers that result in a whole number and perform the same operations until we get 1 in a division by a prime number. That is, $\dfrac{13}{13}=1$ . The prime numbers that divided 143 exactly will be the prime factors of 143.

Complete step by step answer:
We have to find the prime factors of 143. Let us recollect what prime factor is.
Prime factors are factors of a number that are, themselves, prime numbers. We can find the prime factors of a number using prime factorization methods.
Now, consider 143. Let us divide this by the smallest prime number, that is, 2.
$\dfrac{143}{2}=72\dfrac{1}{2}$
We obtained a whole number. Hence we cannot divide 143 exactly by 2.
Let us now consider the next prime number, that is 3. Let’s divide 143 by 3.
 $\dfrac{143}{3}=47\dfrac{2}{3}$
We obtained a whole number. Hence we cannot divide 143 exactly by 3.
Let’s consider the next prime number, that is 5. Let’s divide 143 by 5.
$\dfrac{143}{5}=28\dfrac{3}{5}$
We obtained a whole number. Hence we cannot divide 143 exactly by 5.
Now, we can consider the next prime number, that is 7. Let’s divide 143 by 7.
$\dfrac{143}{7}=20\dfrac{3}{7}$
This is also a whole number. Hence we cannot divide 143 exactly by 7.
Next prime number is 11. When we divide 143 by 11, we will get
$\dfrac{143}{11}=13$
We obtained a prime number. Now let’s divide this result, that is, 13 by the other prime numbers beginning with 2. We will get a whole number for 2,5,7 and 11. But with 13, we will get
$\dfrac{13}{13}=1$
Hence, the prime factors of 143 are $11\times 13$ .

So, the correct answer is “Option B”.

Note: We can also use other methods to find the prime factors of 143. These are shown below.
We can split 143 as $11\times 13$ .
Now, we will find the prime factors of 11 and 13. Since, 11 and 13 are prime numbers, we can skip this step.
Hence, the prime factors of 143 are $11\times 13$ .
Another method is to use the factor tree.
\[\begin{align}
  & \begin{matrix}
   {} & 143 & {} \\
\end{matrix} \\
 & \begin{matrix}
   {} & \swarrow \searrow & {} \\
\end{matrix} \\
 & \begin{matrix}
   {} & 11 & 13 \\
\end{matrix} \\
\end{align}\]
We will split 143 and find the prime factors of the result. This is similar to the previous step except that the factorization is illustrated in a tree model.