What's special about number 3?

Number Theory Level 4

m n p × n p m × p m n = 3 m n p \Large \color{#3D99F6}{m}^{\color{#D61F06}{n}^{\color{#456461}{p}}} \times \color{#D61F06}{n}^{\color{#456461}{p}^{\color{#3D99F6}{m}}} \times \color{#456461}{p}^{\color{#3D99F6}{m}^{\color{#D61F06}{n}}} = \color{#EC7300}3\color{#3D99F6}{m}\color{#D61F06}{n}\color{#456461}{p}

Given that m , n , p \color{#3D99F6}{m}, \color{#D61F06}{n}, \color{#456461}{p} are positive integers that satisfy the equation above, what is the value of m + n + p \color{#3D99F6}{m} + \color{#D61F06}{n} + \color{#456461}{p} ?


The answer is 6.

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Akhil Bansal by Akhil Bansal , 22, India

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2 solutions

Akhil Bansal
Oct 18, 2015

We have, m n p ⋅ n p m ⋅ p m n = 3 m n p m^{n^p}\cdot n^{p^m}\cdot p^{m^n} = 3mnp

⇒ m n p − 1 ⋅ n p m − 1 ⋅ p m n − 1 = 3 \Rightarrow m^{n^p - 1} \cdot n^{p^m - 1}\cdot p^{m^n - 1} = 3

Since m, n, p are +ve integers and 3 is a prime number it follows that

either m n p − 1 = 3 , n p m − 1 = 1 , p m n − 1 = 1 m^{n^p - 1} = 3 , n^{p^m - 1} = 1 , p^{m^n - 1} = 1

⇒ m = 3 , n = 2 , p = 1 \Rightarrow m = 3 , n = 2 , p = 1

or m n p − 1 = 1 , n p m − 1 = 3 , p m n − 1 = 1 m^{n^p - 1} = 1 , n^{p^m - 1} = 3 , p^{m^n - 1} = 1

⇒ m = 1 , n = 3 , p = 2 \Rightarrow m = 1 , n = 3 , p = 2

or m n p − 1 = 1 , n p m − 1 = 1 , p m n − 1 = 3 m^{n^p - 1} = 1 , n^{p^m - 1} = 1 , p^{m^n - 1} = 3 ⇒ m = 2 , n = 1 , p = 3 \Rightarrow m = 2 , n = 1 , p = 3

In either cases , m + n + p = 6 m + n + p = \boxed{6}

20 Helpful 0 Interesting 0 Brilliant 0 Confused

wow u are truly a genius

Kaustubh Miglani - 5 years, 7 months ago

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Thanks, but it took me more than 30 minutes to solves this..

Akhil Bansal - 5 years, 7 months ago

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I did by the same method.

Sarthak Singla - 5 years, 7 months ago

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@Sarthak Singla – hey buddy whats ur score in ftre i know the answer key is wrong but still tell me by the uncorrected answer key.Its out check it. mine is 750 not good at all

Kaustubh Miglani - 5 years, 5 months ago

Did the exact same . Solved it within a minute . Good question

Aditya Kumar - 5 years ago

same method.

brian allen - 3 years, 11 months ago

S i n c e 3 i s a f a c t o r , o n e o f t h e m m u s t b e = 3. A n y o n e c a n b e 3. W L O G m = 3. 3 m = 3 2 . S o o n e o f t h e r e m a i n i n g m u s t b e 2. W L O G n = 2. p = 1 , f o r e q u a t i o n t o b e c o r r e c t . m + n + p = 3 + 2 + 1 = 6. Since~3~is~a~factor,~one~of~them~must~be~=~3.\\ Any~one~can~be~3.~~ WLOG~m=3.\\ 3m=3^2.~So~one~of~the~remaining~must~be~2.~~WLOG~n=2.\\ p=1, ~for~equation~to~be~correct.\\ m+n+p=3+2+1=\color{#D61F06}{6}.

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