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  • Let N = 10800. Then the ratio of number of odd divisors to the number of even divisors isOption: 1 1:2<div class='q

Let N = 10800. Then the ratio of number of odd divisors to the number of even divisors is

Option: 1

1:2


Option: 2

1:3


Option: 3

1:1


Option: 4

1:4


Answers (1)

best_answer

Here N = 24.33.52

To find the number of divisors, we’ll simply use the formula (4 + 1)(3 + 1)(2 + 1) = 60.

To form an even factor, we must select at least one ‘2’ from the lot, which will ensure that whatever be the remaining selection, their multiplication will always result in an even factor.

The number of ways to select at least one ‘2’ from a lot of four identical ‘2’s will be 4 

And, we’ll select any number of ‘3’s and ‘5’s, in 4 and 3 ways respectively.

The required number of ways will, therefore, be 4 x 4 x 3 = 48.

To count the odd factors, we’ll get rid of the ‘2’s. We’ll make the selection from the ‘3’s and the ‘5’s only. The number of selections (or factors) will therefore be (3 + 1)(2 + 1) = 12.

Note that this could also have been obtained by subtracting the even factors from the total, i.e. 60 – 48, which will give the same answer.

The ratio of the number of odd divisors and number of even divisors is: 

\text{Ratio}=\frac{12}{48}=\frac{1}{4}

Posted by

Deependra Verma

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