Question

Find the HCF of $96$ and $404$ by prime factorization method. Also find their LCM.

Answer 1

Hint: HCF (highest common factor) is the greatest number which divides the given numbers completely while LCM (least common multiple) is the smallest number of the common multiples of the given numbers.

Complete step-by-step answer:
We have to find HCF and LCM of $96$ and $404$ using a prime factorization method.
1.To find HCF,
First we find the prime factors of each of the given numbers. Now we know the smallest prime number is $2$ so first we will divide the numbers by it and continue dividing the obtained number again by $2$ till it cannot be divided by it or we get $1$ .If the number is then not divisible by $2$, try the next prime number which will divide it completely.
So the prime factors are written as-
Prime factors of $96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3$
And Prime factors $404 = 2 \times 2 \times 101$
Now identify the common prime factors of the given numbers. Here $2$ is the common factor of $96$ and $2$ is also the common factor of $404$. The common prime factors of the given numbers are-$2 \times 2$
Now multiply all the common factors to find the HCF of the given numbers.
HCF=$2 \times 2 = 4$
2.To find LCM,
First we have to find the prime factors of each number individually and as we already having the prime factors so-
Prime factors of $96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3$
And Prime factors of $404 = 2 \times 2 \times 101$
Now we have to find the number of each prime factor occurring maximum number of times.
So here $2$ occurs $5$ times, $3$ occurs one time and $101$ also occurs one time.
LCM=${2^5} \times 3 \times 101$
Now multiply the numbers to get LCM
LCM=$9696$
Hence answer is HCF of $96$ and $404$ =$4$ and LCM is $9696$.

Note: Prime factorization method is very important in cryptography which is the study of secret codes. Since these secret codes are based on numbers, so to break them we find prime factors of the large numbers as factoring large numbers is very hard and takes a computer a long time to do it.