Question

If $a,b,c$ are prime numbers, $x$ is an even number, $y$ is an odd number. Which of the following are never true?
I. $a + x = b$
II. $b + y = c$
III. $ab = c$
IV. $a + b = c$

Answer 1

Hint:In the above question we can see three terms, prime numbers, even numbers and odd numbers.So, firstly we consider the definitions of all the three terms. Then we will proceed according to the four given statements, using the definitions of the given terms.

Complete step-by-step answer:
We are given that $a,b,c$ are prime numbers ,$x$ is an even number and $y$ is an odd number.
Now firstly let’s consider the definition of prime numbers-
A number that can be divided exactly only by itself and one is called a prime number.
Now definition of even numbers is-
A number that is divisible by two is called an even number.
Also, the definition of odd numbers is-
A number that is not divisible by two is called an odd number.
Now we are given four statements and we have to find the statements which can never be true.
Therefore, to solve the question we will check one by one if the statements are true or not.
Taking statement $I$
$a + x = b$,
Now using the given information, we can write
Sum of a prime number and an even number is equal to a prime number.
Now let’s consider an example
Take
Prime number, $a = 17$,
Even number, $x = 2$
Now taking their sum we get
$
  a + x = 17 + 2 \\
  a + x = 19 \\
 $
And we know $19$ is a prime number.
Hence, we can say that-
Sum of a prime number and an even number is equal to a prime number.
Therefore, $a + x = b$ is true for some values of $a,x,b$.
Hence the statement (1) is true for some values of $a,x,b$.

Now taking Statement $II$
$b + y = c$
Now using the given information, we can write that-
Sum of a prime number and an odd number is equal to a prime number.
Now let’s consider an example
Take
Prime number, $b = 2$
Odd number, $y = 3$
Now taking their sum we get
$b + y = 2 + 3$
$b + y = 5$
And we know $5$ is a prime number.
Hence, we can say that-
Sum of a prime number and an odd number is equal to a prime number.
Therefore, $b + y = c$ is true for some values of $b,y,c$.
Hence the statement (2) is true for some values of $b,y,c$.

Now taking statement $III$
$ab = c$
Now using the given information, we can write that-
Product of two prime numbers is equal to a prime number.
Now, by the definition of prime numbers we know that-
A prime number is a number which is divisible by one or itself.
Now, directly from the definition of prime numbers we can see that the product of two numbers will not be the only divisible by one and itself only.
So, we can say that a product of two prime numbers can never be a prime number.
Hence statement (3) can never be true.

Now taking statement $IV$
$a + b = c$
We can also write it as
Sum of two prime numbers is equal to a prime number.
Now let’s take an example
Prime number, $a = 2$
Prime number, $b = 3$
Now taking their sum we get-
$a + b = 2 + 3$
$a + b = 5$
Now, $5$ is a prime number.
Hence, $a + b = c$ is true for some values of $a,b,c$
From the above conclusion we can write that only statement (III) can never be true.

So, the correct answer is “Statement III”.

Note:In this question, students sometimes get confused about what we have to find. On analyzing the question we can clearly see that we have to find the statement which can never be true means if a statement is true for some values of $a,b,c,x,y$, then it is true but if the statement is not true in any case then we will say that it can never be true. And our answer will be those statements which can never be true.
And to solve the above question basic understanding about prime, even and odd numbers is very important. And we need to think for some examples to prove that the statement is true in some cases.