JEE Mains · Maths · STD 11 - 6. permutation and combination
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 \(6x\)
- D \(12\)
Answer & Solution
Correct Answer
(D) \(12\)
Step-by-step Solution
Detailed explanation
\(y+z=5\) \(\frac{1}{y}+\frac{1}{z}=\frac{5}{6} \quad y > z\) \(\Rightarrow y=3, z=2\) \(\Rightarrow n =2^{ x } \cdot 3^{3} \cdot 5^{2}=(2.2 .2 \ldots)(3.3 .3)(5.5)\) Number of odd divisors \(=4 \times 3=12\)
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