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9^p+27^q+3^5r=3^7s

got it @sauravshakya

What was that?

we just square and cube the value

The question is to find minimum value of p+q+r+s

then common 3 as a base

and then add powers

then you solve it

I have never seen that rule.

ok so go through wolframalpha

I dont think that will also help here.

go through

There is nothing.

I am seeing it.

So, what is the answer?

Give a min, but at tell u the answer can't be find the above way.

What I figured out till now is s must be an odd number....

@experimentX PLZ see this

Cant we find that......

exponential diophantine equations are awful!! you should probably tag mukushla

But he is not online.

you don't do diophantine equations on 8th grade.

Even I dont know that.

This surely cant be a 8th grade question.

@mathslover i think he do this

very dhinchach question

I dont know it even has a solution.

*now ..
seeya later!!

ok...... BYE thanx for trying.

call it hardly a try .. hehehe

@ujjwal p,q,r,s are only positive integers..

@shubhamsrg
( 3^(2p) + 3^(3q) + 3^(5r) )/3 >= 3^( (2p + 3q + 5r)/3 )
HOW???

from AM>=GM

Oh >= got it.

Yeah nice one.

and continuing that,
we have
21s >= 2p + 3q + 5r + 3
but still no idea if that'll help..

And I got:
|dw:1349955495301:dw|

I guess NOW, we can prove it has no SOLUTION

hmm ??

*

THAT GIVES:
|dw:1349955687180:dw|

Now, |dw:1349955794330:dw|

Now, can I say No solution...... Or I lack something.

am sorry,,you assumed 5r>3q ..

but conclusion to my saying is that you made an assumtion..

oh...... no assumtion.

L.H.S or R.H.S can be both negative

Just looking for integer.......
L.H.S will always be integer
R.H.S will always be in decimal.

LHS is not necessary an integer.. we only know that p,q,r,s are integers..

and agreed that RHS will always be non-integer..

Oh ya....... got it @shubhamsrg

So,from there we conclude that 5r<3q

guess not,,how can you say that ?

Because if 5r>=3q then R.H.S will always be integer.

I mean L.H.S

oh wait........ I guess R.H.S can be integer too.

Ok further simplification of it gives:
|dw:1349956922192:dw|

GREAT.

Now, we need to find its minimum value.

what is that fraction?

p+q+r+s = (207p+15)/105

right?

ahh yeah :)

Now, we need to find its minimum integer value.

Oh wait...... But that can make other numbers decimal.

if u wanna check the answer i found is 106 for
\[m=6\]\[p=45\]\[q=30\]\[r=18\]\[s=13\]