Four people think of a number, and then proceed to each tell you something about it.
- A: The number has 2 digits
- B: The number is a divisor of 150
- C: The number is not 150
- D: The number is divisible by 25
Unfortunately, one of these people is not telling the truth. Who is it?
(In reply to Answer
by K Sengupta)
Let us respectively denote the number by n and the no. of digits
contained in it by d.
Assume that A is lying. Is so, then it follows that d! = 2. in terms
of D's true statement, it follows that d>=3, while by B's statement it follows that d<=3. This is possible iff d=3. However, by the respective statements of C and D, it follows that n!= 150 and n divides 150. Now, a 3 digit number cannot divide 150, unless it is equal to 150. This is a contradiction. Therefore, A is telling the truth.
Assume that B is the liar. if so, it follows that 25 divides n,
n does not divide 150 and d=2. the two digit multiples of 25
are 25, 50 and 75- each of which divide 150. This is a contradiction. Therefore, B is telling the truth.
Assume that C is the liar. Then, it follows that n=150, giving d=3, which contradicts A's true statement that d=2. Therefore, C is telling the truth.
Assume that D is the liar. Then, it follows that d=2, n divides 150, n!=150 and 25 does not divide n. There are three numbers that simultaneously satisfy the four restrictions, and these are given by n = 10, 15, 30. This is indeed in conformity with the provisions of the given problem.
Consequently, D is the individual who is lying and the number
correspond to precisely one amongst 10, 15 and 30.