Perfect Square, Perfect Cube
Find the smallest positive integer which is both a perfect square and a perfect cube.
The answer is 1.
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15 solutions
Then is 1^0=1?
why not 729? 729=27 27=9 9*9
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Are square & cube of 729 equal ?
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dear we need only a single digit like (a=b=1) not a=9 & b=27...plz tell me if u find other than 1...best of luck in solving ur answer
no but 729 is both a perfect square as well as a perfect cube 729=9^3=27^2 so N=729 a=9 b=27 1 is the most obvious answer .
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@Madhavan Rajagopal – sry abt that mistook the problem
It doesn't mean that they shud be equal... The question is that to find the smallest number which is perfect square and cube as well.
What's the smallest such number?
It's asking for the smallest number...
64 is the second smallest. 64=4^3=8^2, so in every cases 729 is not the answer
In Google calculator, 99*9 says it's 891. 729=9^3=27^2, so it's 3rd smallest.
How about 0? 0 is also natural number, right?
Question says positive integer. 0 is neither positive nor negative
no. its not. 0 is a whole number.
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natural numbers are not always defined the same, thats why you should use Z > 0
The number in question must be a square and a cube so it must be a number that is a perfect sixth power. That is, we're looking for a number in the form n 6 , where n is a natural number. Since we're looking we have to find the smallest possible natural natural number in this form, we'll use the smallest natural number n , which is 1. Therefore, our answer will be 1 6 = 1 .
It has been proven rather recently ( i.e. about 2004 ) that there are NO nontrivial solutions the equation n^2 = m^3. By trivial, we mean n = 1 and n = 0. Going even further, this very difficult proof shows that the equation n^2 - n^3 = 1 has just one nontrivial solution, and that solution is 3^2 - 2^3 = 9 - 8 = 1. As an indicator of how difficult this proof is, it was found a number of years after Andrew Wiles proved Fermat's Last Theorem.
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No non-trivial solutions to n^2=m^3??? Any (n,m) of the form (k^3,k^2) is a solution.
As for the second thing, n^2= m^3+1, this is an olympiad question. Stop saying random shit.
x^2=x^3 then x^3-x^2=O then x(x-1)=O Then ans is x=1
Doesn't work for numbers like 64 however.
I thought this at first as well, however this just proves zero and 1 are the numbers that you can square and cube to get the same value.
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No plz try N=729 =3^6 6th power of any integer exhibits such behaviour
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729 is a perfect cube and a perfect square. Of course anything of the form n 6 can be written as ( n 3 ) 2 or ( n 2 ) 3 . However I was responding to the problem x 2 = x 3 . Which is saying "What number when squared is equal to it's cubed version." Of which the solution is 0 and 1. 0 2 = 0 = 0 3 and 1 2 = 1 = 1 3 . However ( n 6 ) 2 = n 1 2 = n 1 8 = ( n 6 ) 3 Now the point I was trying to make is that what you said was correct, which is why the way Alind said would not work.
Zero is not a natural number
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zero is a natural number
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@Youssef Hassan F – its not. 0 is a whole number.
its 1 but it might be zero too
By brute force method!
a=1 b=1 N=1
a=8 b=4 N=64
a=27 b=9 N=729
a=64 b=16 N=4096
a=125 b=25 N=15625
a=216 b=36 N=46656
in short 6th power of any integer shows this property
for any integer N M=N^6=((N)^3)^2=((N)^2)^3 N=1 M=1 N=2 M=64=((2)^3)^2=((2)^2)^3 N=3 M=729=((3)^3)^2=((3)^2)^3 N=4 M=4096=64^2=16^3 and it goes on....
1 is the first natural number and it happens to be a perfect square, a perfect cube, a perfect fourth power, and so on. In short, is always 1, so it is the smallest natural number that is both a perfect square and a perfect cube.
The answer is 1. because it is the only number which has a perfect square and a perfect cube.
Well it's not the only, just the smallest
64 is also a number which is a perfect square and a perfect cube 4×4×4=64 8×8=64 So one is not the only such number..... I m sorry to say that your answer is correct but your concepts are wrong !!!
64 is also a perfect square and perfect cube But the question asked for the smallest number ..
I first thought of 0 and then 64 and then 1 came to my mind
because 0 is not natural number
1 is a perfect square and a perfect cube
1 squared is 1, 1 cubed is 1. The smallest: 1 64 729 4096 15625 and so on.
Lets say x is the answer. x= z^2 = y^3 we assume that x = 2n (2n)^3 - (2n)^2 = 0 8n^3 - 4n^2 = 0 4n^2 (2n - 1) = 0 n=0 or n= 1/2
Back to the first statement that x = 2n. n= 1/2, then x = 1
At first I thought it was 64, but then it said incorrect and I then knew it had to be smaller. Eventually I got 1.
1 is the first natural number and it is a square,a cube,a fourth power...In short it is itself raised to n so the answer is 1
1^n=1. This makes it 1 whatever the power
1 is the first natural number and it happens to be a perfect square, a perfect cube, a perfect fourth power, and so on. In short, 1 n is always 1, so it is the smallest natural number that is both a perfect square and a perfect cube.