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If r, s and t are consecutive integers, what is the greatest prime fac

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Bunuel
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If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   07 Mar 2017, 11:54
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If r, s and t are consecutive integers, what is the greatest prime factor of 3^r + 3^s + 3^t ?

A) 3
B) 5
C) 7
D) 11
E) 13
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E
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   07 Mar 2017, 12:09
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Let R = x, S = x+1, T = x+2

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

Therefore greatest prime factor of 3^r + 3^s + 3^t is 13

Answer is E) 13
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   07 Mar 2017, 13:42
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Let’s pick smart numbers.

Let r =0, let s= 1 and let t= 2

Hence the equation becomes 3^0+ 3^1 + 3 ^2

Which is equal to 1+3+9

= 13 = 13*1

Therefore the greatest prime factor is 13. Hence E.
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   09 Mar 2017, 14:48
As they are consequative we can write it so: 3^r(1+3^1+3^2)=3^r*13
So answer is 13 (E)
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   15 Mar 2017, 16:40
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Bunuel wrote:
If r, s and t are consecutive integers, what is the greatest prime factor of 3^r + 3^s + 3^t ?

A) 3
B) 5
C) 7
D) 11
E) 13


Let’s say r = 1, s = 2, and t = 3; then we have:

3^1 + 3^2 + 3^3 = 3^1(1 + 3 + 9) = 3(13)

Let’s say r = 2, s = 3, and t = 4; then we have:

3^2 + 3^3 + 3^4 = 3^2(1 + 3 + 9) = 3^2(13)

We see that regardless of the 3 consecutive positive integers we select, the largest prime will always be 13 since the other factor is always a power of 3.

Answer: E
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink] New post   03 Feb 2023, 00:58
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Bunuel wrote:
If r, s and t are consecutive integers, what is the greatest prime factor of 3^r + 3^s + 3^t ?

A) 3
B) 5
C) 7
D) 11
E) 13


Solution:

  • Questions like this is perfect for number picking
  • Especially because the answer options are all in whole number and none of them are none of the above or cannot be determined

  • So, let us assume \(r=0\), \(s=1\) and \(t=2\)
  • So, \(3^r + 3^s + 3^t=3^0+3^1+3^2=1+3+9=13\)
  • So, the greatest prime factor of \(3^r + 3^s + 3^t\) is 13

Hence the right answer is Option E
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Re: If r, s and t are consecutive integers, what is the greatest prime fac [#permalink]
03 Feb 2023, 00:58
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