Example 1. Prove that there are exactly four numbers, each of three digits, having the following properties: (a) all the three digits are distinct (b) 7 divides the number (c) the number obtained by reversing the order of the number is also divisible by 7.
Example 2. Each of the three girls Sheela, Leela and Sameera, has in her purse exactly one of the following objects: a pencil, a ball-pen and an eraser. Out of the following statements one is true and two are false: (i) Sheela has a pencil (ii) Leela does not have the pencil (iii) Sameera does not have the eraser. Determine who has what object.
Of course, you can find the solution just below, but it is highly recommended that you first try to solve it on your own.
Just remember the words of Paul Halmos, who says “the only way to learn mathematics is to do mathematics”.
Solution 1. Let the three-digit number be xyz, where x, y, z are all distinct, 0 ≤ x, y, z ≤ 9 and x ≠ 0.
This number can be written as 100x + 10 y + z = (98 + 2)x + (7 + 3)y + z = 7(14x + y) + (2x + 3y + z).
If this number is divisible by 7, then (2x + 3y + z) must be a multiple of 7.
The number obtained by reversing the digits is
zyx = 100z + 10y + x = 7(14z + y) + (2z + 3y + x).
Since this number is also divisible by 7, (2z + 3y + x) must be a multiple of 7. Since (2x + 3y + z) and (2z + 3y + x) are both multiples of 7, therefore (2x + 3y + z) − (2z + 3y + x) = (x – z) is also a multiple of 7. Since x ≠ z and 0 ≤ x, z ≤ 9, only possibility is x – z = ±7.
Suppose x > z. Then x = z + 7. The possible values of x and z are (x, z) = (7, 0), (8, 1), (9, 2). Now (x, z) = (7, 0) together with (2x + 3y + z) be a multiple of 7 gives y = 0 and 7. None of them are acceptable as the value of y.
Similarly, (x, z) = (8, 1) together with (2x + 3y + z) be a multiple of 7 gives y = 6. This time, the number is xyz = 861.
Finally, (x, z) = (9, 2) together with (2x + 3y + z) be a multiple of 7 gives y = 5. This time, the number is xyz = 952.
Similarly considering x < z, we get two more numbers 168 and 259 (just by reversing the previously obtained numbers), which also satisfy the given condition.
Solution 2. Suppose statement (i) is true. Then Sheela has a pencil, and therefore, Leela does not have the pencil. This means that statement (ii) is also true. However, both (i) and (ii) cannot be true. So, statement (i) is false.
Assume that (ii) is true. Then, Leela does not have the pencil. Consequently, Sameera has the pencil, and therefore statement (iii) is true. Since both statements (ii) and (iii) cannot be true, therefore statement (ii) is false.
Since statements (i) and (ii) are both false, therefore (iii) must be true. This means that Smaeera does not have either the eraser or the pencil. This implies that Sameera has the ball pen.
Since Leela has the pencil and and Sameera has the ball pen, therefore Sheela has the eraser.
Hence, Sheela has the eraser, Leela has the pencil and Sameera has the ball pen.