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Prime number is simply that number which is only divisible by itself and ofcourse by 1 also. Like 17, Is there any number which can divide 17 completely? It can be only divisible by 1 and 17 itself (17/1 =17 , 17/17 = 1) hence it is prime no.
Co prime- It is always set of numbers which doesnt have any common factor except 1.
Example : let us take 15 and 8
15 = 1*3*5
8 = 1*2*2*2
only 1 is common factor in 15 and 8 hence they are co-prime to each other
A prime number is defined as a number which has no factor other than 1 and itself. But, coprime numbers are considered in pairs and two numbers are coprime if they have a common factor as 1 only.