Correct Option (2)
The problem states that PQR, PPT are 3-digit numbers and PS is a 2-digit number, where P, Q, R, S, T are distinct non-zero digits. The given equation is PQR − PS = PPT. Additionally, Q = 3 and T < 6.
Let's analyze the subtraction in column format:
P Q R - P S ------- P P T
From the units column:
- For the units digit of the result to be T, and given that Q=3, a borrow from the tens place (Q) is necessary because if R ≥ S, then Q-P would yield P, which is not possible for Q=3.
- Thus, (10 + R) − S = T. This implies S − R = 10 − T.
From the tens column:
- Since a borrow occurred from Q, the tens digit becomes (Q − 1).
- So, (Q − 1) − P = P.
- Substituting Q = 3: (3 − 1) − P = P.
- 2 − P = P.
- 2 = 2P, which gives P = 1.
From the hundreds column:
- P − 0 = P. This is consistent with P = 1.
Summary of derived conditions:
- P = 1
- Q = 3 (given)
- S − R = 10 − T
- P, Q, R, S, T must be distinct non-zero digits.
- T < 6 (given)
Considering the distinct non-zero digits constraint, P=1 and Q=3 are already used. The possible values for T (T < 6 and T ≠ 1, T ≠ 3) are 2, 4, 5.
Let's examine each possible value of T:
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Case 1: T = 2
- Digits used: {P=1, Q=3, T=2}.
- Equation: S − R = 10 − 2 = 8.
- Available digits for R, S (excluding 1, 2, 3): {4, 5, 6, 7, 8, 9}.
- To satisfy S − R = 8, with R and S from the available set, no valid pair exists. The maximum difference in this set is 9 − 4 = 5.
- Number of (R, S) pairs: 0.
-
Case 2: T = 4
- Digits used: {P=1, Q=3, T=4}.
- Equation: S − R = 10 − 4 = 6.
- Available digits for R, S (excluding 1, 3, 4): {2, 5, 6, 7, 8, 9}.
- Possible pairs (R, S) such that S − R = 6:
- If R = 2, then S = 2 + 6 = 8. The digits {1, 3, 2, 8, 4} are all distinct and non-zero. This is a valid pair: (2, 8).
- No other value of R from the available set yields a valid S.
- Number of (R, S) pairs: 1.
-
Case 3: T = 5
- Digits used: {P=1, Q=3, T=5}.
- Equation: S − R = 10 − 5 = 5.
- Available digits for R, S (excluding 1, 3, 5): {2, 4, 6, 7, 8, 9}.
- Possible pairs (R, S) such that S − R = 5:
- If R = 2, then S = 2 + 5 = 7. The digits {1, 3, 2, 7, 5} are all distinct and non-zero. This is a valid pair: (2, 7).
- If R = 4, then S = 4 + 5 = 9. The digits {1, 3, 4, 9, 5} are all distinct and non-zero. This is a valid pair: (4, 9).
- No other value of R from the available set yields a valid S.
- Number of (R, S) pairs: 2.
Total number of possible distinct (R, S) pairs = 0 (for T=2) + 1 (for T=4) + 2 (for T=5) = 3.
Incorrect Options:
Options 1, 3, and 4 are incorrect because the detailed analysis of possible values for T (2, 4, 5) and the subsequent derivation of valid (R, S) pairs, while adhering to all given constraints (distinct non-zero digits, P=1, Q=3, S − R = 10 − T), yields exactly 3 such pairs. Therefore, any other count is inaccurate.
