Question

If we have the sets $aN=\\{ax/x\in N\\}$ and $bN\cap cN=dN$ Where $b,c\in N$ are relatively prime then
1. $d=bc$
2. $c=bd$
3. $b=cd$
4. $a=bd$

Answer 1

To solve this problem, first with the help of given set try to find out the set of $bN$ and $cN$ , after that try to find out the intersection of both the sets and then according to the given condition in the question, try to simplify further and you will get your required answer.

Complete step-by-step solution:
Prime numbers can be defined as the positive integers that have only two factors, $1$ and the integer itself. In other words we can also say that the prime numbers can be defined as the numbers, which are only divisible by $1$ or the number itself. The prime number was discovered by Eratosthenes. The prime numbers that have only one composite number between them are called the twin prime numbers or twin primes (In other words we can also say that the pair of prime numbers that differ by $2$ only are known as twin prime numbers.
Two numbers are said to be relatively prime when they have only $1$ as the common factor or we can say that there is no same value other than that, you could divide them both and get zero as the remainder. If we want to find whether the two numbers are relative prime or not, then we need to find the HCF of both the numbers and if the HCF is $1$ , then two numbers are said to be relatively primes otherwise not.
Now according to the given question:
$bN=\\{bx/x\in N\\}$ Can be defined as the set that consists of all positive integer multiples of $b$
$cN=\\{cx/x\in N\\}$ Can be defined as the set that consists of all positive integer multiples of $c$
And we are given that: $bN\cap cN$
It can be defined as the set that consists of positive integers that are multiples of both $b$ and $c$
$\Rightarrow bc=bcN$
Since, $b$ and $c$ are prime.
As, $bN\cap cN=dN$
$\Rightarrow bcN=dN$
$\Rightarrow bc=d$
Hence, the correct option from all the above options is $1$.

Note: Some properties of prime numbers are as: Any number that is greater than $1$ can be divided by at least one prime number, Any even positive integer greater than $2$ can be expressed as the sum of two prime numbers and except $2$ all other prime numbers are odd.