Question

What is the number of even factors of 36000 which are divisible by 9 but not
by 36​

Answer 1

Step-by-step explanation:

Factorise 3600 :

3600 = 2 x 2 x 2 x 2 x 3 x 3 x 5 X 5

= 2⁴ x 3² x 5²

No of factors = (4 + 1)(2 + 1)(2 + 1) = 45

So far so good.

But why on earth we add 1 to the powers of distinct prime factors ?

There is a very interesting property of exponents. Any number raised to the power 0 will always produce 1.

Therefore, we can write the above number as :

3600 = 2⁴ x 2^0 x 3² x 3^0 x 5² x 5^0 instead of 3600 = 2⁴ x 3² x 5²

Hence, we have one additional power for each distinct prime factors that is hidden in normal course !!!!

Using same argument, we can restate 3600 as :

3600 = 2⁴ x 2^0 x 3² x 3^0 x 5² x 5^0 x 7^0 x 11^0 x 13^0 x 17^0 x 19^0 X 23^0 ……..

All prime factors other than 2,3 & 5 have only nuisance value and do not change the product in anyway.

Number of factors become (4+1)(2+1)(2+1)(0+1)(0+1)(0+1)(0+1)(0+1)…..

= 45