Question

Show that 5309 and 3072 are prime to each other.

Answer 1

Hint: Here, we are required to show that the given two numbers are prime to each other. Will we find the factors of these two numbers and then compare them to find their Highest Common Factor or HCF. If the HCF of these two numbers turns out to be 1, then, these two numbers are co-prime or in other words, prime to each other. There should be no number other than 1 which is common between the given two numbers.

Complete step-by-step answer:
Prime numbers are those numbers which have only 2 factors. This means that when we try to find the factors of prime numbers then we can get only 2 factors wherein, one of those 2 factors is the number itself and the second factor is 1.
Now, we have to find whether 5309 and 3072 are prime to each other or not.
First of all, we will find the factors of both the numbers.
As we can observe, the number 5309 does not have any factors at all except 1 and 5309 itself. Hence, this is a perfect example of a prime number.
Therefore, the factors of 5309 are:
 \[5309 = 1 \times 5309\]
Now, when we try to find the factors of 3072:
 \[\begin{array}{l}1 \times 3072 = 3072\\2 \times 1536 = 3072\\3 \times 1024 = 3072\\4 \times 768 = 3072\\6 \times 512 = 3072\\8 \times 384 = 3072\\12 \times 256 = 3072\\16 \times 192 = 3072\\24 \times 128 = 3072\\32 \times 96 = 3072\\48 \times 64 = 3072\end{array}\]
There are 21 factors of 3072 and they are- 1,2,3,4,6,8,12,16,24,32,48,64,96,128,192,256,384,512,768,1024 and 1536.
So, if we try to find out the highest common factor or HCF of these numbers then, leaving 1, no other number is common between the two.
Hence, 5309 and 3072 are prime to each other or in other words, they are co-prime numbers.

Note: A number can be defined as either prime or composite number. The numbers which have more than 2 factors are called composite numbers. The smallest prime number is 2 as its factors are 1 and the number 2 itself. Also, 1 is the only number which is neither prime nor composite. If we talk about co-prime numbers, then, two numbers are co-prime if their highest common factor or HCF is 1. This means that the highest and the greatest common factor between them is 1.