Q55·CSAT · Prelims 2024
Finding the number of factors of a large composite number
NumericalFactors & MultiplesFactual single● Medium
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?
Options
a
3
b
4
cAnswer
5
d
6
Explanation
Test consecutive values of k to find when unique identification fails under the constraint p < q for positive integers:
If k = 3: p+q=3 \implies only pair is (1, 2). (Unique)
If k = 4: p+q=4 \implies only pair is (1, 3) since p cannot equal q. (Unique)
If k = 5: p+q=5 \implies valid pairs are ⟨MATH⟩(1, 4)⟨/MATH⟩ and ⟨MATH⟩(2, 3)⟨/MATH⟩.
Since k=5 yields two completely different integer combinations that satisfy all rules, it is the lowest value where uniqueness is lost.
For partitioned integer sums where p < q, the threshold for generating multiple solutions occurs at k = 5 for positive integers.
Answer: (c).
Question details
Year
2024
Paper
CSAT
Question
Q55
Section
Numerical Ability
Sub-topic
Factors & Multiples
Type
Factual single
Difficulty
Medium
Source hint
Number theory divisor counting
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