Let p and q be positive integers satisfying p

Question

Let p and q be positive integers satisfying p<q and p+q=k. What is the smallest value of k that does not determine p and q uniquely?

Accepted Answer

The correct answer is option C. We are looking for the smallest positive integer k such that the equation p+q=k, with the conditions p and q are positive integers and p<q, yields more than one distinct pair for (p, q).

Let's test values of k:
- If k = 3: The only pair (p,q) satisfying p+q=3 and p<q (with p,q positive integers) is (1, 2). This is unique.
- If k = 4: The only pair (p,q) satisfying p+q=4 and p<q (with p,q positive integers) is (1, 3). This is unique.
- If k = 5: The possible pairs (p,q) satisfying p+q=5 and p<q (with p,q positive integers) are:
    - (1, 4)
    - (2, 3)
  Since there are two distinct pairs for k=5, the values of p and q are not uniquely determined.

Therefore, the smallest value of k that does not determine p and q uniquely is 5. This question tests basic number theory and logical enumeration.

Q55. Let p and q be positive integers satisfying p<q and p+q=k. What is the smallest value of k that does not determine p and q uniquely?

Detailed Solution

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