- A. 3
- B. 4
- C. 5
- D. 6
Answer: B
Explanation
We are given a 5-digit number PQRST where P, Q, R, S, T are distinct digits.
1. T ≠ 0
2. P = 3T
3. S = Q + 4
4. Q = R + 3
From (3) and (4), we can express S and Q in terms of R:
Q = R + 3
S = (R + 3) + 4 = R + 7
Now let’s consider possible values for T, keeping in mind P=3T and P must be a single digit (P ≠ 0 as it’s the first digit of a 5-digit number):
* If T = 1, then P = 3 × 1 = 3.
Digits used so far: P=3, T=1. (Distinct)
Now consider R. R must be a digit distinct from 1 and 3.
* If R = 0: Q = 0 + 3 = 3, S = 0 + 7 = 7. Digits: P=3, Q=3, R=0, S=7, T=1. Here Q=P, which violates the ‘distinct digits’ condition. (Invalid)
* If R = 2: Q = 2 + 3 = 5, S = 2 + 7 = 9. Digits: P=3, Q=5, R=2, S=9, T=1. All are distinct. Number: 35291. (Valid)
* If T = 2, then P = 3 × 2 = 6.
Digits used so far: P=6, T=2. (Distinct)
Possible values for R (distinct from 2 and 6):
* If R = 0: Q = 0 + 3 = 3, S = 0 + 7 = 7. Digits: P=6, Q=3, R=0, S=7, T=2. All are distinct. Number: 63072. (Valid)
* If R = 1: Q = 1 + 3 = 4, S = 1 + 7 = 8. Digits: P=6, Q=4, R=1, S=8, T=2. All are distinct. Number: 64182. (Valid)
* If T = 3, then P = 3 × 3 = 9.
Digits used so far: P=9, T=3. (Distinct)
Possible values for R (distinct from 3 and 9):
* If R = 0: Q = 0 + 3 = 3, S = 0 + 7 = 7. Digits: P=9, Q=3, R=0, S=7, T=3. Here Q=T, which violates the ‘distinct digits’ condition. (Invalid)
* If R = 1: Q = 1 + 3 = 4, S = 1 + 7 = 8. Digits: P=9, Q=4, R=1, S=8, T=3. All are distinct. Number: 94183. (Valid)
* If T = 4, then P = 3 × 4 = 12. P must be a single digit, so T cannot be 4 or greater.
The possible 5-digit numbers are: 35291, 63072, 64182, 94183. There are 4 such numbers. This question tests logical deduction and systematic enumeration of possibilities, common in number puzzles.