Fundamental Drill#2

Number Theory Level 3

Determine whether the statements below are true or false.

Statement 1:

If the digit $8$ is added at the end of a positive integer divisible by $8$ , the new number must be divisible by $8$ too.

Statement 2:

If a positive integer divisible by $8$ and ends with digit $8$ has its end digit removed. The remaining number must be divisible by $8$ too.

Note: The original number should not be single-digit.

Give your answer according to the respective order of the statements.

You can try more of my fundamental problems here .

True,False False,True Both are true. Both are false.

1 solution

Donglin Loo
Jun 20, 2018

$\textbf{Statement 1:}$

Let the original number be $x$

$x$ is divisible by $8$

$\therefore x\equiv0(mod8)$

$\Rightarrow 10x\equiv0(mod8)$

$\Rightarrow 10x+8\equiv0(m0d8)$

$10x+8$ is divisible by $8$

$\therefore$ the new number formed must be divisible by $8$ .

$\therefore$ $\textbf{Statement 1}$ is $\boxed{true}$

$\textbf{Statement 2:}$

Let the original number be $10y+8$ .

$10y+8$ is divisible by $8$ .

$\therefore 10y+8\equiv0(mod8)$

$\Rightarrow 10y\equiv0(mod8)$

$\textbf{But}$

this does not indicate that $y$ must be $\equiv0(mod8)$

For example, when $y=4$

$10y=40\equiv0(mod8)$

But $y$ is not divisible by $8$

$\therefore$ $\textbf{Statement 2}$ is $\boxed{false}$