Question 53

The smallest positive integer n with 24 divisors considering 1 and n as divisors is

Solution

For any given number, that can be represented as $$ A^{x} \times B ^ {y} $$, etc

The number of factors is denoted by (x+1) x ( y+1), etc

360 = $$ 2 ^ {3} \times 3 ^ {2} \times 5^ {1}$$

So the number of factors = (3 +1) x (2+ 1) (1+ 1) = 4x3x2 = 24

For 240, it is $$ 2 ^ {4} \times 3 ^ {1} \times 5^ {1}$$

Number of factors = 5 x 2 x 2 = 20 only

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