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LCM of Polynomials

The smallest ...

Question

The smallest positive integer n' with 24 divisors (where 1 and n are also considered as divisors of n) is

420

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240

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360

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480

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Solution

The correct option is C 360
$L e t u s i n v e s t i g a t e t h e o p t i o n s a s p e r t h e g u i d e l i n e s o f t h e g i v e n p a s s a g e . O p t i o n A ⟶ 420 = 2^{2} \times 3^{1} \times 23^{1} \Rightarrow a = 2, b = 1, c = 1 ∴ n u m b e r o f d i v i s o r s = (2 + 1) (1 + 1) (1 + 1) = 12 \neq 24. O p t i o n B ⟶ 240 = 2^{4} \times 3^{1} \times 5^{1} \Rightarrow a = 4, b = 1, c = 1 ∴ n u m b e r o f d i v i s o r s = (4 + 1) (1 + 1) (1 + 1) = 20 \neq 24. O p t i o n C ⟶ 360 = 2^{3} \times 3^{2} \times 5^{1} \Rightarrow a = 3, b = 2, c = 1 ∴ n u m b e r o f d i v i s o r s = (3 + 1) (2 + 1) (1 + 1) = 24. O p t i o n D ⟶ 480 = 2^{5} \times 3^{1} \times 5^{1} \Rightarrow a = 5, b = 1, c = 1 ∴ n u m b e r o f d i v i s o r s = (5 + 1) (1 + 1) (1 + 1) = 24. B o t h o p t i o n C a n d D c o m p l i e s w i t h t h e g i v e n g u i d e n c e . B u t o p t i o n C i . e 360 i s l e s s e r . ∴ A n s w e r - O p t i o n C$

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