Question

if a is number of all even divisors and b is number of all odd divisors for 10800 then 2a+3b is. options 1 . 72. 2. 132. 3. 96. 4. 136​

Answer 1

2a+3b = 132

Step-by-step explanation:

We are given that a is number of all even divisors and b is number of all odd divisors for 10800.

For this, we will do prime factorization of the number 10800.

10800 = 2 $\times 5400$

5400 = 2 $\times$ 2700

2700 = 2 $\times$ 1350

1350 = 2 $\times$ 675

675 = 3 $\times$ 225

225 = 3 $\times$ 75

75 = 3 $\times$ 25

25 = 5 $\times$ 5

5 = 5 $\times$ 1

Now, we achieved 1 as the last quotient so will stop the procedure now.

So, prime factorization of 10800 =  $2^{4}\times 3^{3}\times 5^{2}$

Total number of divisors = (4+1) $\times$ (3+1) $\times$ (2+1) = 60

Number of odd divisors = a = (1 case of $2^{0}$) $\times$ (4 cases of $3^{0}, 3^{1}, 3^{2}, 3^{3}$) $\times$ (3 case of $5^{0}, 5^{1}, 5^{2}$)

                                      = 1 $\times$ 4 $\times$3 = 12

{Here, $2^{1}, 2^{2}, 2^{3} \text{and} 2^{4}$ is not taken because this will make them even factors}

So, number of even divisors = b =  60 - 12 = 48.

Now, 2a + 3b = (2 $\times$ 48) + (3 $\times$ 12)

                       =  96 + 36 = 132.