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Number Theory Level 1

Which is the smallest $natural$ number which is a $perfect$ $square$ as well as a $perfect$ $cube$ ?

Hint : Name of the problem !

Enjoy Math !

The answer is 1.

5 solutions

Parth Panchal
May 1, 2015

Solve it like this

x=x^2

Therefore, x=1;x Belongs to N ------------------------>(1)

Even Further

x^2=x^3

Therefore , x=1;x belongs to N ------------------------>(2)

Thus From (1)&(2)

The answer is 1

Karandeep Singh Ludhar
Mar 10, 2015

Let x be the number . Then according to question x^3 = x^2 : x^3 - x^2 = 0
x^2 (x - 1) = 0 : x - 1 = 0 ;therefore; x =1

You must specify that $x \neq 0$ before dividing $x^2$ from both sides.

Nihar Mahajan - 6 years, 3 months ago

But is it required to prove this , since the question asks for a natural number .

karandeep singh ludhar - 6 years, 3 months ago

You must just mention it.

Nihar Mahajan - 6 years, 3 months ago

@Nihar Mahajan – Okay , THANKS!!!!!!!!!

karandeep singh ludhar - 6 years, 3 months ago

Its not required to mention $x\neq 0$ as the question clearly asks for a $natural$ number but its not a crime if we mention though !!!!!

Gagan Raj - 6 years, 3 months ago

Whenever you divide a number by a variable it 'must' be mentioned that the variable is not equal to 0. Since the question asks for natural numbers , 'HENCE' $x\neq 0$ .

Nihar Mahajan - 6 years, 3 months ago

@Nihar Mahajan – Just out of curiosity i'm asking this........can you explain the statement "Whenever you divide a number by a variable it 'must' be mentioned that the variable is not equal to 0" ?????

Can you tell me what is the $number$ and what is the $variable$ in this question ?????

Gagan Raj - 6 years, 3 months ago

@Gagan Raj – Its obvious! in the above solution , $x$ is the variable , $0$ is the number.

Nihar Mahajan - 6 years, 3 months ago

Sidh Satam
Jul 11, 2015

Can someone explain the so called "hint" that is mentioned? I got the answer but I have no idea what it has got anything to do with the hint given

Noel Lo
May 27, 2015

For extra insight, a perfect square must be expressible in exponential form where the base is an integer while the power is divisible by 2. Likewise, a perfect cube must be expressible in a form where the base is an integer while the power is divisible by 3. Therefore, for a number to be both a perfect square and cube at the same time, it must be expressible in a form where the base is an integer while the power is divisible by both 2 and 3, in other words divisible by 6. The first few integers to satisfy this are 1, 64, 729...

Gagan Raj
Mar 9, 2015

The smallest natural number which satisfies all the given conditions is in fact the smallest natural number $\boxed{1}$ .

The solution provided by karandeep is not really correct. What Karandeep wrote is the answer to the question: "what is the smallest natural number which is the square and cube of the same number". 1 is the only solution to this problem (not just the smallest). With regards to the problem posed though 64 is also a solution and so is 81 and so on... (although they are not the smallest). Hope u all c the difference.

Sandipan Chowdhury - 6 years, 2 months ago

Winner !

The answer is 1.

5 solutions

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