Consider the following statements: I. There exists a natural number which when increased by 50% can have its…
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?
I only
II only
Both I and II
Neither I nor II
Solution
Statement I: increasing by 50% multiplies by $1.5 = 3/2$. Take $N=2$: factors of 2 are {1,2} (2 factors); $2\times1.5 = 3$ has factors {1,3} (2 factors) - unchanged. So I is correct. Statement II: increasing by 150% multiplies by $2.5 = 5/2$. Take $N=2$: $2\times2.5 = 5$, factors {1,5} (2 factors) - same as 2. So II is correct. Both statements are correct.