How many three-digit numbers are possible such that the difference between the original number and the number obtained by reversing the digits is 396? (no digit is repeated)
The number obtained by reversing the digits is 100c + 10b + a.
The difference between the original number and the reversed number is given as 396.
(100a + 10b + c) – (100c + 10b + a) = 396
99a – 99c = 396
99(a – c) = 396
a – c = 396 / 99 = 4.
1. a is a digit from 1 to 9.
2. c is a digit from 0 to 9.
3. b is a digit from 0 to 9.
4. a, b, c are distinct (a != b, b != c, a != c).
5. a – c = 4.
Since a – c = 4 and a is a single digit, a > c, which guarantees a != c. Also, since a >= 1, c >= 0.
Let’s list the possible pairs of (a, c) where a – c = 4 and a is in {1..9}, c is in {0..9}:
– If c = 0, a = 4. Pair (4, 0).
– If c = 1, a = 5. Pair (5, 1).
– If c = 2, a = 6. Pair (6, 2).
– If c = 3, a = 7. Pair (7, 3).
– If c = 4, a = 8. Pair (8, 4).
– If c = 5, a = 9. Pair (9, 5).
There are 6 such pairs for (a, c).
For each pair (a, c), the digit b must be distinct from a and c. There are 10 possible digits (0-9). Since a and c are distinct and are already chosen, b can be any of the remaining 10 – 2 = 8 digits.
If we strictly follow N – N_rev = 396, there are 6 * 8 = 48 such numbers. However, 48 is not among the options.
Let’s consider a common convention in such problems: the reversed number must also be a three-digit number. This implies that the units digit of the original number, c, cannot be 0.
If c must be in {1..9} (and a in {1..9}) with a-c=4:
– If c = 1, a = 5. Pair (5, 1).
– If c = 2, a = 6. Pair (6, 2).
– If c = 3, a = 7. Pair (7, 3).
– If c = 4, a = 8. Pair (8, 4).
– If c = 5, a = 9. Pair (9, 5).
There are 5 such pairs for (a, c) if c!=0.
For each of these 5 pairs, b must be distinct from a and c. There are 10 – 2 = 8 possible digits for b.
Total number of such three-digit numbers = 5 pairs * 8 options for b per pair = 40.
This matches option D. This suggests the implicit condition that the reversed number is also a three-digit number (c!=0) was intended.