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How many factors of 24 × 35 × 104 are perfect squares which are greater than 1?

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Calculation:

We get on prime factorizing

⇒ 24 × 35 × 24 × 54

⇒ 28 × 35 × 54

We get the perfect squares factors which is greater than 1, when the power of factor is even.

For the prime number 2, 

We get 20, 22, 24, 26, 28  there are 5 cases

For the prime number 3, 

We get 30, 32, 3 there are 3 cases

For the prime number 5, 

We get 50, 52, 5 there are 3 cases

The total number of factors having even power = 5 × 3 × 3 = 45

But, there is 1 case, we get 2× 30  ×  5= 1

The factor is equal to 1, so we need to eliminate it because we need a factor greater than 1

The total number of factors having even power which is greater than 1 = 45 - 1 = 44

∴ The total number of factors having even power which is greater than 1 is 44.

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