prince is thinking about a six digit number.The sum of the digits of the number is 43.If only two statement is true then find the number.
statements:a)it is a perfect square,b)it is a perfect cube,c)it is less than 500000
prince number
 Thanic Nur Samin
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 Joined: Sun Dec 01, 2013 11:02 am
Re: prince number
$\lfloor\sqrt{500000}\rfloor=707$
$707^2=499849$
$499849=707^2, 499849<500000, 4+9+9+8+4+9=43$
$707^2=499849$
$499849=707^2, 499849<500000, 4+9+9+8+4+9=43$
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Re: prince number
What is the sign $\left \lfloor \right \rfloor$ means ???
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Re: prince number
There are three cases to consider . Here's two of them .
1. First assume statement 'a' and 'b' is correct .
Since $ [2,3] = 1$ from the statements it can be said that the number is a perfect sixth power . Now ,
$ \sqrt [6]{100000} > 6$ and $ \sqrt [6]{999999} < 10$ .
So , the number would be a sixth power of $7$ , $8$ or $9$. But , $ 7^6 = 117649$ , $ 8^6 = 262144$ and $ 9^6 = 531441$ where the digit sum is not $43$ . So, there isn't any such number for this case .
2. Now assume statement 'b' and 'c' is correct .
We know that dividing a number and its digit sum separately by $ 9$ leave the same remainder . So , by statement 'b' and 'c' , the number is congruent to $7$ modulo $9$ . But every perfect cube is congruent to $ {{1 , 0 , 1}}$ modulo $9$ . Hence there's no solution in this one too .
But I can't find any logical approach for the third case (when assuming statement 'a' and 'b' is correct) .
1. First assume statement 'a' and 'b' is correct .
Since $ [2,3] = 1$ from the statements it can be said that the number is a perfect sixth power . Now ,
$ \sqrt [6]{100000} > 6$ and $ \sqrt [6]{999999} < 10$ .
So , the number would be a sixth power of $7$ , $8$ or $9$. But , $ 7^6 = 117649$ , $ 8^6 = 262144$ and $ 9^6 = 531441$ where the digit sum is not $43$ . So, there isn't any such number for this case .
2. Now assume statement 'b' and 'c' is correct .
We know that dividing a number and its digit sum separately by $ 9$ leave the same remainder . So , by statement 'b' and 'c' , the number is congruent to $7$ modulo $9$ . But every perfect cube is congruent to $ {{1 , 0 , 1}}$ modulo $9$ . Hence there's no solution in this one too .
But I can't find any logical approach for the third case (when assuming statement 'a' and 'b' is correct) .
Last edited by sadman sakib on Sat Mar 08, 2014 4:30 pm, edited 2 times in total.
Re: prince number
He said only two of the statements are true, so it's not necessary that the number is a perfect sixth power.sadman sakib wrote:Since $ [2,3] = 1$ from the statements it can be said that the number is a perfect sixth power . Now ,
$ \sqrt [6]{100000} > 6$ and $ \sqrt [6]{500000} < 9$ .
So , the number would be a sixth power of $7$ or $8$ . But , $ 7^6 = 117649$ and $ 8^6 = 262144$ where the digit sum is not $43$ . So, there isn't any such number .
 What is the value of the contour integral around Western Europe?
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
Re: prince number
It's called the "floor" function.tanmoy wrote:What is the sign $\left \lfloor \right \rfloor$ means ???
The floor of a number is the highest integer less than or equal to it.
Thus, $\left \lfloor 2.9 \right \rfloor = 2$, $\left \lfloor 2.9 \right \rfloor = 3$, $\left \lfloor 5 \right \rfloor = 5$.
See this article for information.
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Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth."  Sherlock Holmes