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Q.

Find the sun of the digits of the least natural number $P$, such that the sum of the cubes of the four smallest distinct divisors of $P$ equals $2P$.

 A.

7

 B.

8

 C.

9

 D.

10

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Solution:
Option(C) is correct

Let the least number be $P$, 1 is its least divisor.
Let $2^{nd}, 3^{rd}$ and $4^{th}$ least divisors be $x,y$ and $z$ respectively.

We consider the following values of divisor $a$ and the corresponding values of $a^3$, from $x,y$ and $z$ exactly 1 or all 3 are odd.($P$ is even)

$a=1 : a^3 = 1$
$a=2 : a^3 = 8$
$a=3 : a^3 = 27$
$a=4 : a^3 = 64$
$a=5 : a^3 =125$
$a=6 : a^3 = 216$

For $x, y$ and $z = (2,3,4)$, $2P = 100$ (i.e. $P=50$). But 3 is not a divisor of 50.

For$ x,y,z = (2,3,6)$, $2P = 252$ (i.e. $P = 126$) and the $1,2,3,6$ are four least distinct divisor of 126.

The required number is 126. The sum of digits is 9.


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