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How many factors of 2^95 are there which are greater than 1,000,000?

 

I tried several things but I couldn't find the right answer that made sense

 


 

 May 31, 2019
 #1
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39614081257132168796771975168 = 2^95 (95 prime factors, 1 distinct)

 

It has 96 DIVISORS (factors) and 76 of them are over 1,000,000.  You obtain them as follows:

 

2^0, 2^1, 2^2, 2^3, 2^4...........and so on until you get to: 2^20 = 1,048,576. So, the powers of 2 from 0 to 19 =20 factors(divisors) that are under 1,000,000.

 May 31, 2019
edited by Guest  May 31, 2019
 #2
avatar+1081 
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I'm sorry, I wasn't allowed to use a calculator for this type of problem.

 

I knew that at some point 2^n was going to be over 1,000,000.

 

Is there a method to get there without using a calculator?

 

The problem can be super easy after you find out 2^n is larger than 1,000,000

 May 31, 2019
edited by CalculatorUser  May 31, 2019
 #3
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I suppose you could do it in your head by knowing that 2^10 is just over 1,000. So, 2^20 would be 1,000 x 1,000 +.

The only factors it has are 2s. You can multiply 2 by itself ANY number of times up to 95 times and the results will all be factors of 95.

 May 31, 2019
 #4
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Yes, that makes sense.

 

Because 2^10 is slightly over 1000

 

So 2^20 is kinda slightly over 1,000,000

 

We know that 2^19 is NOT over 1,000,000 because 2^20 divided by 2 will be significantly under 1,000,000

 May 31, 2019
 #5
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Factors of 2^95 = {1,2,4,8,16,...,2^94,2^95}.

The first power of 2 that is greater than 1,000,000 is 2^20, which is 1.048,576.

Number of factors of 2^95 which are greater than 1,000,000 = 95 - 20 + 1 = 76

 Jun 1, 2019

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