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# prime factors ps

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Director
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prime factors ps [#permalink]  08 Jan 2006, 21:50
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If r, s, t are consecutive integers, what is the greatest prime factor of 3^r +3^s + 3^t ?
VP
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Is it 13?

Let r = k,
s = k+1
t = k +2

N = 3^k + 3^(k+1) + 3^(k+2)
= 3^k (1 + 3 + 9)
= 3^k * 13

The only prime factors of N are 3 and 13.
So 13 is the highest prime.
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giddi77 wrote:
Is it 13?

Let r = k,
s = k+1
t = k +2

N = 3^k + 3^(k+1) + 3^(k+2)
= 3^k (1 + 3 + 9)
= 3^k * 13

The only prime factors of N are 3 and 13.
So 13 is the highest prime.

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Auge um Auge, Zahn um Zahn !

Director
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oa is 13, can you explain this part? thanks in advance.

giddi77 wrote:
Is it 13?

Let r = k,
s = k+1
t = k +2

N = 3^k + 3^(k+1) + 3^(k+2)
= 3^k (1 + 3 + 9)

= 3^k * 13

The only prime factors of N are 3 and 13.
So 13 is the highest prime.
VP
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Posts: 1066
Location: USA
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joemama142000 wrote:
oa is 13, can you explain this part? thanks in advance.

giddi77 wrote:
Is it 13?

Let r = k,
s = k+1
t = k +2

N = 3^k + 3^(k+1) + 3^(k+2)
= 3^k (1 + 3 + 9)

= 3^k * 13

The only prime factors of N are 3 and 13.
So 13 is the highest prime.

3^(k+1) = 3*3^k = 3*3^k
3^(k+2) = 3^2*3^k = 9*3^k
HTH
_________________

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

- Bernard Edmonds

VP
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13...as above. took the numbers as r, r+1, r+2.
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If r = n, s = n+1, t = n+2

3^r +3^s + 3^t = 3^n + 3^(n+1) + 3^(n+2) = 3^n (1 + 3 + 9) = 13(3^n)

Largest prime factor = 13
Senior Manager
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Can someone just clarify how 3^n(13) = 13 (3^n)????

My brain is hurting!
VP
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13...
Used the same solution as above... 3^n x 13.
CEO
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Can someone just clarify how 3^n(13) = 13 (3^n)????

My brain is hurting!

Because r,s,t are consecutive

Let r = x
s = x+1
t = x+2

Then 3^r + 3^s + 3^t becomes
3^k + 3^(k+1) + 3^(k+2)
= 3^k + 3^k * 3^1 + 3^k * 3^2
= (3^k) * (1+3+9)
= (3^k) * (13)
Hence largest prime factor = 13
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SAID BUSINESS SCHOOL, OXFORD - MBA CLASS OF 2008

Senior Manager
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ps_dahiya,

I just didn't understand the step "3^n(13) = 13 (3^n)" in ywilfred's post.

Does this mean a^b(c) = c(a^n)

I think I may be confusing the terms....does 3^n(13) mean 3 to the power of 13n OR 3 to the power of n, multiplied by 13?

Enlighten me!
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