How many prime factors does 5^40+5^37 have : PS Archive
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# How many prime factors does 5^40+5^37 have

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How many prime factors does 5^40+5^37 have [#permalink]

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06 Jan 2005, 10:50
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How many prime factors does 5^40+5^37 have?
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06 Jan 2005, 11:16
38*2*2*3 - 1

which turns out to be 455.

here is the approach:
5^37(125+1)

5^37*126 = (5^37)*(2^1)*(7^1)*(9^2)

now taking only the powers

(37+1)*(1+1)*(1+1)*(2+1) - 1 = 455
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Bhimsen Joshi

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06 Jan 2005, 11:20
Most of your approach is correct, I think you misunderstand the question in the last step

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06 Jan 2005, 11:26
Oops I agree...I missed the Prime part of the question...
I think I should go to bed now...thanks...
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Bhimsen Joshi

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06 Jan 2005, 11:27
4 is the ans.

5^37(5^3+1) = 5^37(126) = 5^37x2x7x3x3 ===> i.e. 4 distinct prime factors (5, 3, 2, 7).
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06 Jan 2005, 11:40
I do not get either Bhimsen or Banerjeea_98's approach. Can someone show me the light and explain a bit more in details?

Thanks!

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06 Jan 2005, 11:45
[quote="jinino"]I do not get either Bhimsen or Banerjeea_98's approach. Can someone show me the light and explain a bit more in details?

Thanks!

:oops:[/quote]

jinino, 5^40+5^37 can be written as 5^37(5^3+1) which again means 5^37(126). There is only 1 prime factor in 5^37 i.e. 5 and there are 3 prime factors for 126 (2x7x3x3) i.e. 3, 7, 2. Hence ans is 4. Hope this helps.
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06 Jan 2005, 11:46
jinino wrote:
I do not get either Bhimsen or Banerjeea_98's approach. Can someone show me the light and explain a bit more in details?

Thanks!

5^40+ 5^37 = 5^37 * (5^3+1)

5^37 will have only 1 prime factor, which is 5 itself

(5^3+1) = 126, 126 has 3 prime factors, 2, 3, and 7

so total is 4 prime factors

hope it will help
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06 Jan 2005, 12:40
DLMA, banerjeea_98,

Thanks for your input. I got it now, but I still have one more question. We know 5^37 has only one prime number, which is 5. What do we do if the question is 6^40+6^37? How do we figure out the prime number for 6^37?
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Re: PS A lil tricky one [#permalink]

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07 Jan 2005, 04:45
DLMD wrote:
How many prime factors does 5^40+5^37 have?

6^37 = (2*3)^37 = (2^37) * (3^37)

So 6^37 has 2 prime factors: 2 and 3
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07 Jan 2005, 08:55
artabro,

I understand now, thanks!
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