Question

HCF of 24, 36 and 92 is:
A.24
B.36
C.12
D.4

Answer 1

Hint: Use prime factorization and find the highest common factors of all the three numbers. The product of those common factors gives the HCF.

Complete step by step answer:
HCF stands for Highest Common Factor.
Hence, it is the highest number which is a common factor of all the three given numbers. One of the most basic and effective methods to find the HCF can be by using the prime factorization method.
In this method, we start with the smallest prime number that divides the given number and divide the number by it. Then we divide the result by the smallest prime number by which it is divisible, and repeat the process till the result is 1.
Using this method, the prime factorization of 24 will be:
$24=2\times 12$ (Since, 2 is the smallest prime number by which 24 is divisible)
Now, we repeat the same for the result, i.e., 12.
$12=2\times 6$
$6=2\times 3$
And, \[3=3\times 1\]
Hence, we arrive at the result 1, and so we stop.
Now, all these prime factors of 24 (numbers in blue) are written and this becomes the prime factorization of 24
$24=2\times 2\times 2\times 3$
Now we will see the prime factorization of 36
\[\begin{align}
  & 36=2\times 18 \\
 & 18=2\times 9 \\
 & 9=3\times 3 \\
 & 3=3\times 1 \\
\end{align}\]
Hence, the prime factorization of 36 is
$36=2\times 2\times 3\times 3$
Similarly, the prime factorization of 92
\[\begin{align}
  & 92=2\times 46 \\
 & 46=2\times 23 \\
 & 23=23\times 1 \\
\end{align}\]
Hence, the prime factorization of 92 is
$92=2\times 2\times 23$
We now find the common factors amongst these three.
$\begin{align}
  & 24=2\times 2\times 2\times 3 \\
 & 36=2\times 2\times 3\times 3 \\
 & 92=2\times 2\times 23 \\
\end{align}$
Thus, we see that the first two numbers $2\times 2$ are common in each of three prime factorizations.
Hence, the product of these, 4 is the required Highest Common Factor of the three given numbers.
Thus, the correct answer is 4, option (d).

Note:
A short-cut method to solve the same question is by dividing the three numbers in the question by each of the options and checking for the option that divides all the three given numbers.
For option (a), only 24 is divisible by 24; and the remaining two numbers, 36 and 92 are not. Hence it is not the correct answer.
For option (b), only 36 is divisible by 36; and the remaining two numbers, 24 and 92 are not. Hence it is also not the correct answer.
For option (c), only 24 and 36 are divisible by 12; and 92 is not. Hence it is also not the correct answer.
For option (d), all the three numbers 24, 36 and 92 are divisible by 4. Hence it is correct.