Let pp, qq and rr be 2-digit numbers where p< q< r . If pp + qq + rr = tt0 , where tt0 is a 3-digit number ending with zero, consider the following statements: 1. The number of possible values of p is 5. 2. The number of possible values of q is 6. Which of the above statements is/are correct?
- A1 only
- B2 only
- CBoth 1 and 2Correct
- DNeither 1 nor 2
Explanation
The problem involves interpreting the notation 'pp', 'qq', 'rr', and 'tt0'.
- 'tt0' is a 3-digit number ending with zero. This implies it's a multiple of 10.
- The sum is pp + qq + rr = tt0. If 'p' were a 2-digit number (e.g., p=12), then 'pp' would typically mean 1212 (a 4-digit number). In this case, the sum of three 4-digit numbers would be a 4-digit or 5-digit number, not a 3-digit number 'tt0'. Therefore, the only way for the equation to hold and 'tt0' to be a 3-digit number is if 'p', 'q', 'r' are single-digit numbers, and 'pp' means 11*p (e.g., if p=1, pp=11; if p=5, pp=55). The phrase "p, q, r be 2-digit numbers" is a contradiction in the question, and we must proceed with 'p', 'q', 'r' being single digits for a consistent solution.
Let p, q, r be single-digit numbers (1-9). Given p = 3: Smallest q is p+1, smallest r is q+1. So p+q+r >= 3+4+5 = 12 > 10. No solutions.
Case 2: S = p + q + r = 20 We need p = 6: Smallest sum is 6+7+8 = 21 > 20. No solutions.
Now let's evaluate the statements:
Statement 1: The number of possible values of p is 5. From Case 1 (S=10), possible values for p are {1, 2}. From Case 2 (S=20), possible values for p are {3, 4, 5}. Combining these, the set of possible values for p is {1, 2, 3, 4, 5}. The number of possible values of p is 5. Statement 1 is CORRECT.
Statement 2: The number of possible values of q is 6. From Case 1 (S=10): (1, 2, 7) -> q=2 (1, 3, 6) -> q=3 (1, 4, 5) -> q=4 (2, 3, 5) -> q=3 The possible values for q are {2, 3, 4}.
From Case 2 (S=20): (3, 8, 9) -> q=8 (4, 7, 9) -> q=7 (5, 6, 9) -> q=6 (5, 7, 8) -> q=7 The possible values for q are {6, 7, 8}.
Combining these, the set of possible values for q is {2, 3, 4, 6, 7, 8}. The number of possible values of q is 6. Statement 2 is CORRECT.
Since both statements are correct, the answer is C.
The final answer is C) Both 1 and 2
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