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 $n$, including 1 , is :

• A. 11
• B. 6
• C. 6 x
• D. 12

Answer 1

Answer: (D)

$y+z=5$
$\frac{1}{y}+\frac{1}{z}=\frac{5}{6} \,\,\,\, y>z$
$\Rightarrow y =3, z =2$
$\Rightarrow n =2^{ x } \cdot 3^{3} \cdot 5^{2}=(2.2 .2 \ldots)(3.3 .3)(5.5)$
Number of odd divisors $=4 \times 3=12$