This is because ac = a * c, i.e., by the definition of the | (divides) relation.MI5 said:It's not so clear to me why c|ac.
I still don't understand I'm afraid. Could you say bit more please?Evgeny.Makarov said:This is because ac = a * c, i.e., by the definition of the | (divides) relation.
I need to be sure you know the definition of the relation denoted by |. Could you write this definition?MI5 said:I still don't understand I'm afraid. Could you say bit more please?
The statement "c|ab" means that c is a common factor of both a and b, while the statement "gcd(b, c) = 1" means that b and c have no common factors other than 1. This suggests that c is a prime factor of a, since it is the only factor that is common between a and b.
The condition gcd(b, c) = 1 is necessary in order to show that c is a prime factor of a. If gcd(b, c) was not equal to 1, then c would not be a prime factor of a and the statement would not hold true.
Yes, for example, if a = 12, b = 3, and c = 4, then c|ab is true since 4 is a factor of 12 and 3, and gcd(b, c) = 1 is also true since the only common factor of 3 and 4 is 1.
Yes, the statement is always true. This is because if c is a factor of both ab and ac, then it must also be a factor of a (since c is a common factor of both ab and ac, it can be factored out of both terms leaving c multiplied by a). This is known as the transitive property of divisibility.
This statement is closely related to the concept of prime factorization, which states that every positive integer can be expressed as a unique product of prime numbers. Since gcd(b, c) = 1 in this statement, it shows that c is a prime factor of a, which aligns with the concept of prime factorization.