JEE PYQ: Permutation And Combination Question 13
Question 13 - 2021 (26 Feb Shift 2)
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:
(1) 11
(2) 6x
(3) 12
(4) 6
Show Answer
Answer: (3)
Solution
$y + z = 5$ and $\frac{1}{y} + \frac{1}{z} = \frac{5}{6} \Rightarrow \frac{y+z}{yz} = \frac{5}{6} \Rightarrow yz = 6$. So $y = 3, z = 2$. Odd divisors (ignore $2^x$): $(y+1)(z+1) = 4 \times 3 = 12$.
BETA
