## prince number

For students of class 6-8 (age 12 to 14)
Sidharth
Posts: 20
Joined: Sat Feb 08, 2014 11:29 pm
Location: Naogaon,Bangladesh

### prince number

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

Thanic Nur Samin
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

### Re: prince number

$\lfloor\sqrt{500000}\rfloor=707$
$707^2=499849$
$499849=707^2, 499849<500000, 4+9+9+8+4+9=43$
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tanmoy
Posts: 289
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### Re: prince number

What is the sign $\left \lfloor \right \rfloor$ means ???
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sadman sakib
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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) .
Last edited by sadman sakib on Sat Mar 08, 2014 4:30 pm, edited 2 times in total.

Nirjhor
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### Re: prince number

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 .
He said only two of the statements are true, so it's not necessary that the number is a perfect sixth power.
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Labib
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Location: Dhaka, Bangladesh.

### Re: prince number

tanmoy wrote:What is the sign $\left \lfloor \right \rfloor$ means ???
It's called the "floor" function.
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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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes