UPSC Prelims 2025 CSAT, question 64

Topic: Number system

Q64

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 ?

Solution.

Answer: (c) Both I and II

Statement I is correct: increasing $N = 12$ ( $6$ factors: $1, 2, 3, 4, 6, 12$ ) by $50\%$ gives $1.5 \times 12 = 18$ ( $6$ factors: $1, 2, 3, 6, 9, 18$ ), leaving the number of factors unchanged.
Statement II is correct: increasing $N = 20$ ( $6$ factors: $1, 2, 4, 5, 10, 20$ ) by $150\%$ gives $2.5 \times 20 = 50$ ( $6$ factors: $1, 2, 5, 10, 25, 50$ ), also leaving the number of factors unchanged.