Natural numbers with unchanged factor count
Question
Consider the following statements: I. There exists a natural number which when increased by 50% can have its number of factors unchanged. II. There exists a natural number which when increased by 150% can have its number of factors unchanged. Which of the statements given above is/are correct?
- 1.
There exists a natural number which when increased by 50% can have its number of factors unchanged.
- 2.
There exists a natural number which when increased by 150% can have its number of factors unchanged.
Options
I only
II only
Both I and II
Neither I nor II
Explanation
We test small natural numbers to verify existence proofs. Statement I: Increasing N by 50\% means multiplying by 1.5. To yield a natural number, N must be even. Let N=2. The factors of 2 are {1, 2} (total 2 factors). Increasing by 50\% gives 2 × 1.5 = 3. The factors of 3 are {1, 3} (total 2 factors). Since 2=2, Statement I is true. Statement II: Increasing N by 150\% means multiplying by 2.5. Again, let N=2. 2 × 2.5 = 5. The factors of 2 are {1, 2} (total 2). The factors of 5 are {1, 5} (total 2). Statement II is also true.
Answer: (c).
Question details
Year
2025
Paper
CSAT
Question
Q74
Section
Numerical Ability
Sub-topic
Factors & Divisors
Type
Number theory
Difficulty
Hard
Source hint
Number theory
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