A natural number has

Question:

A natural number has prime factorization given by $n=2^{x} 3^{y} 5^{z}$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}, y>z$. Then the number of odd divisors of $\mathrm{n}$, including 1 , is :

  1. 11

  2. 6

  3. 6x

  4. 12


Correct Option: , 4

Solution:

$y+z=5$

$\frac{1}{y}+\frac{1}{z}=\frac{5}{6}$$y>z$

$\Rightarrow y=3, z=2$

$\Rightarrow \mathrm{n}=2^{\mathrm{x}} \cdot 3^{3} \cdot 5^{2}=(2.2 .2 \ldots)$

Number of odd divisors $=4 \times 3=12$

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