Question

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

(a) I and II
(b) II and III
(c) Only III
(d) III and IV

Answer 1

Hint: This is a trick question and the only way to solve this question is to consider various values for a, b, c, x and y according to the given conditions and check for the given four statements. We will try to prove that each statement is true and consider only those values which satisfy the given statements.

Complete step-by-step answer:
First of all, let us consider that a = 2, b = 3, c = 5, x = 4 and y = 7.
a + x = 2 + 4 = 6. This is not equal to b = 3. Therefore, statement I is false with these values.
b + y = 3 + 7 = 10. This is not equal to c = 5. Therefore, statement II is also false with these values.
ab = 2(3) = 6. This is not equal to c = 5. Therefore, statement III is also false with these values.
a + b = 2 + 3 = 5 = c. Thus, statement IV is true with these values.
Therefore, statement IV can be true.
After this, we will not check for statement IV as it is proved that it can be true.
Now, let us consider that a = 3, b = 7, c = 11, x = 4 and y = 5.
a + x = 3 + 4 = 7 = b. Thus, statement I is true.
b + y = 7 + 5 = 12. This is not equal to c = 11. Therefore, statement II is also false with these values.
ab = 3(7) = 21. This is not equal to c = 11. Therefore, statement III is also false with these values.
After this, we will not check for statement I and IV as it is proved that they can be true.
Now, let us consider that a = 7, b = 2, c = 5, x = 4 and y = 3.
b + y = 2 + 3 = 5 = c. Thus, statement II is true with these values.
ab = 7(2) = 14. This is not equal to c = 5. Therefore, statement III is also false with these values.
Thus, statement III is never true.
Therefore, option (c) is the correct option.

Note: This is a very unique question and students have to consider various values of such kinds of questions. For this particular question, students can directly conclude that statement III will never be true as it defies the basic definition of prime numbers.