# Number TheoryShow that the number a is not a square of an integer

#### mathmari

##### Well-known member
MHB Site Helper
Hey! I have to show that the number $a=201340168052123987111222893$ is not a square of an integer, without doing calculations.

Could I solve this in $\mathbb{Z}_8$?
I mean that the number $a$ can be written as followed:
$$a=3+9 \cdot 10 +8 \cdot 10^2 + 2 \cdot 10^3+...$$
Since at $\mathbb{Z}_8$: $[10]=[2], [10^2]=[4], [10^3]=[8]=[0], [10^k]=[0] \text{ for }k \geq 3$ we have:
$$[a]=[3]+[9] \cdot [10]+[8] \cdot [10^2]+[2] \cdot [10^3]+...=[3]+[1] \cdot[2]=[3]+[2]=[3+2]=[5]$$
We suppose that $a$ is a square of an integer, so $a=b^2 \Rightarrow [a]=[b^2]$. So it must be be $[b^2]=[5]$. The possible values of $[b^2]$ are:
$$\left.\begin{matrix} : & [0] & [1] & [2] & [3] & [4] &[5] & [6] & [7]\\ [b^2]:& [0] &[1] & [4] & [1] &[0] & [1] &[4] & [1] \end{matrix}\right.$$
Since there is not the value $[5]$, it cannot be true..So $a$ cannot be a square of an integer.

Is this correct? Or can I not just solve this in $\mathbb{Z}_8$?

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#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hey! I have to show that the number $a=201340168052123987111222893$ is not a square of an integer, without doing calculations.

Could I solve this in $\mathbb{Z}_8$?
I mean that the number $a$ can be written as followed:
$$a=3+9 \cdot 10 +8 \cdot 10^2 + 2 \cdot 10^3+...$$
Since at $\mathbb{Z}_8$: $[10]=[2], [10^2]=[4], [10^3]=[8]=[0], [10^k]=[0] \text{ for }k \geq 3$ we have:
$$[a]=[3]+[9] \cdot [10]+[8] \cdot [10^2]+[2] \cdot [10^3]+...=[3]+[1] \cdot[2]=[3]+[2]=[3+2]=[5]$$
We suppose that $a$ is a square of an integer, so $a=b^2 \Rightarrow [a]=[b^2]$. So it must be be $[b^2]=[5]$. The possible values of $[b^2]$ are:
$$\left.\begin{matrix} : & [0] & [1] & [2] & [3] & [4] &[5] & [6] & [7]\\ [b^2]:& [0] &[1] & [4] & [1] &[0] & [1] &[4] & [1] \end{matrix}\right.$$
Since there is not the value $[5]$, it cannot be true..So $a$ cannot be a square of an integer.

Is this correct? Or can I not just solve this in $\mathbb{Z}_8$?

Yep. All correct!

Note that it is slightly easier in $\mathbb Z_{10}$.
What are the possibilities for the last digit of any square?

#### Deveno

##### Well-known member
MHB Math Scholar
As ILS points out, the last digit (in base 10, our usual base system) of a perfect square must be either:

0,1,4,5,6 or 9.

3 is not on this list.

#### mathmari

##### Well-known member
MHB Site Helper
Yep. All correct!

Note that it is slightly easier in $\mathbb Z_{10}$.
What are the possibilities for the last digit of any square?
As ILS points out, the last digit (in base 10, our usual base system) of a perfect square must be either:

0,1,4,5,6 or 9.

3 is not on this list.