Positive integer not divisible by 7

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January 14th 2011, 06:02 AM
Hellbent

Positive integer not divisible by 7

Hi,

Is there a simple explanation for this?

The positive integer $n$ is not divisible by $7$ . The remainder when $n^2$ is divided by 7 and the remainder when $n$ is divided by $7$ are each equal to $k$ . What is $k$ ?

A. 1
B. 2
C. 4
D. 6
E. It cannot be determined from the information given.

I just looked at the middle part and concluded the answer to be 1. The only numbers I know that are equal to their squares are 0 and 1. Zero isn't available.
January 14th 2011, 06:18 AM
Unknown008

I can't see any answer besides A. ..

Note that it cannot be zero because otherwise, n would be a multiple of 7.
January 14th 2011, 06:27 AM
Pranas

Quote:

Originally Posted by Hellbent

Hi,

Is there a simple explanation for this?

The positive integer $n$ is not divisible by $7$ . The remainder when $n^2$ is divided by 7 and the remainder when $n$ is divided by $7$ are each equal to $k$ . What is $k$ ?

A. 1
B. 2
C. 4
D. 6
E. It cannot be determined from the information given.

I just looked at the middle part and concluded the answer to be 1. The only numbers I know that are equal to their squares are 0 and 1. Zero isn't available.

I have to agree.
Let's say our number can be written as:
$n = 7m + k$
Therefore ${n^2} = 49 \cdot {m^2} + 14 \cdot m \cdot k + {k^2} = 7 \cdot (7 \cdot {m^2} + 2 \cdot m \cdot k) + {k^2}$

We actually have to keep in mind, that we are trying to evaluate $k$ itself and only remainder of ${k^2}$ when it's divided by 7.
However, in your case there really are no other possible scenarios other than $k = {k^2} = 1$ (you can run through $k = \overline {1,6}$ to make sure of that).
January 15th 2011, 03:46 AM
Archie Meade

Quote:

Originally Posted by Hellbent

Hi,

Is there a simple explanation for this?

The positive integer $n$ is not divisible by $7$ . The remainder when $n^2$ is divided by 7 and the remainder when $n$ is divided by $7$ are each equal to $k$ . What is $k$ ?

A. 1
B. 2
C. 4
D. 6
E. It cannot be determined from the information given.

I just looked at the middle part and concluded the answer to be 1. The only numbers I know that are equal to their squares are 0 and 1. Zero isn't available.

$\displaystyle\frac{n^2}{7}=m+\frac{k}{7}\Rightarro w\ n^2-7m-k=0$

$\displaystyle\frac{n}{7}=x+\frac{k}{7}\Rightarrow\ n-7x=k$

Substitute k

$n^2-7m-n+7x=0\Rightarrow\ n^2-n+7(x-m)=0$

$\Rightarrow\ (n+7)(n+x-m)=n^2+(-1)n+7(x-m)\Rightarrow\ x-m=-8\Rightarrow\ m-x=8$

$(n-8)(n+7)=0\Rightarrow\ n=8$

$\displaystyle\frac{n^2}{7}=\frac{64}{7}=9\frac{1}{ 7}$

$\displaystyle\frac{8}{7}=1\frac{1}{7}$

The remainder is 1.
January 15th 2011, 04:37 AM
Pranas

Quote:

Originally Posted by Archie Meade

$\displaystyle\frac{n^2}{7}=m+\frac{k}{7}\Rightarro w\ n^2-7m-k=0$

$\displaystyle\frac{n}{7}=x+\frac{k}{7}\Rightarrow\ n-7x=k$

Substitute k

$n^2-7m-n+7x=0\Rightarrow\ n^2-n+7(x-m)=0$

$\Rightarrow\ (n+7)(n+x-m)=n^2+(-1)n+7(x-m)\Rightarrow\ x-m=-8\Rightarrow\ m-x=8$

$(n-8)(n+7)=0\Rightarrow\ n=8$

$\displaystyle\frac{n^2}{7}=\frac{64}{7}=9\frac{1}{ 7}$

$\displaystyle\frac{8}{7}=1\frac{1}{7}$

The remainder is 1.

Could you please clarify how did you come up with
$\displaystyle \((n+7)(n+x-m)=n^2+(-1)n+7(x-m)\Rightarrow\ x-m=-8\Rightarrow\ m-x=8$

and $\displaystyle \[{\rm{n = 8}}\]$

If you concluded, that there is only one appropriate value for $\displaystyle \[{n}}\]$ under given circumstances, that is not true.
Otherwise my apologies if I misunderstood something.
January 15th 2011, 04:51 AM
Archie Meade

Quote:

Originally Posted by Pranas

Could you please clarify how did you come up with
$\displaystyle \((n+7)(n+x-m)=n^2+(-1)n+7(x-m)\Rightarrow\ x-m=-8\Rightarrow\ m-x=8$

and $\displaystyle \[{\rm{n = 8}}\]$

If you concluded, that there is only one appropriate value for $\displaystyle \[{n}}\]$ under given circumstances, that is not true.
Otherwise my apologies if I misunderstood something.

Of course, multiples of 7 may be added to n=8 for a full solution.
My earlier post outlines a "beginning" for another analysis.