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If n = 10800 then find the total number of divisors of n and the number of even divisors

60, 48
48, 60
32, 60
60, 32

hint Hint:

We are given a number 10800. We have to find the number of divisors it has. and number of even divisiors it has. Divisior is a number that divides the given number giving zero reminder. We will start with finding the prime factors of the number. We will their power to solve the question.

The correct answer is: 60, 48

The given number is 10800. We will do following steps to find the answer.
Step 1: Prime factorisation
Prime factors means we write the given number in terms of multipliction of smallest prime numbers.
As the given number is even, we will start with 2.
10800 = 2 × 5400
= 2 × 2 × 2700
= 2 × 2 × 2 × 1350
= 2 × 2 × 2 × 2 × 675
= 2 × 2 × 2 × 2 × 3 × 225
= 2 × 2 × 2 × 2 × 3 × 3 × 75
= 2 × 2 × 2 × 2 × 3 × 3 × 3 × 25
= 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5
= 2⁴ × 3³ × 5²
Step 2: We will collect the powers of all the factors. We will add 1 in them. Then we will multiply them. The final value is the number of divisors the given number has.
Number of divisors = (4 + 1) × (3 + 1) × (2 + 1)
= 5 × 4 × 3
= 60
So, total number of divisors is 60.
Step 3: We will collect the powers of odd factors. We will add them and multiply. We will subtract the value from the total number of divisors to get the even number of divisors.
Odd number of divisors = (3 + 1) × (2 + 1)
= 4 × 3
= 12
Even divisors = Total number of divisors - Odd number of divisors
= 60 - 12
= 48
So, the number of even divisiors is 48.
The right option is 60,48

For such questions, we should know how to find prime factors.

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If n = 10800 then find the total number of divisors of n and the number of even divisors

60, 48 48, 60 32, 60 60, 32

We are given a number 10800. We have to find the number of divisors it has. and number of even divisiors it has. Divisior is a number that divides the given number giving zero reminder. We will start with finding the prime factors of the number. We will their power to solve the question.

The correct answer is: 60, 48

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60, 48
48, 60
32, 60
60, 32

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