Find integers a and b congruent to mod

Question

Can anyone give me a hint of how to solve this equation

Find integers $a$ and $b$ such that $a + b ≡ a - b \bmod 5$.

I did not go over this in my class. Kindly help me on how to solve this one. Thanks

Answer 1

Here's a start: note that what you have written implies that $b=0 \text{ (mod 5)}$. To see this, recall that what it means to be equal to some residue mod 5 is that $a+b = a-b + 5k$ for some $k\in\mathbb{Z}$. Normal arithmetic reveals that $2b=5k$, which means that $b$ has to be divisible by 5. See if you can figure it out from here!

Answer 2

Subtracting $a$ from both sides and adding $b$ gives $$2b\equiv 0\mod{5}$$ $$\therefore b=5k$$ For $k\in \mathbb{Z}$. $a$ can take any value in $\mathbb{Z}$ as the equivalence does not depend on $a$.

Answer 3

You could write the congruence in equation form as follows

$$ a+b\equiv a-b\ \bmod 5 \Rightarrow a+b=a-b+k\cdot 5, \ \ k\in\mathbb{Z} $$

which is equivalent to

$$ 2b=k\cdot 5 \Rightarrow 2b\equiv 0 \ \bmod 5. $$ Now, you can divide here by $2$ since it is relative prime to $5$

so you only need to define $b$ for a such congruence to hold and $b$ should be divisible by $5$ so $b\in\{5, 15, -10 \}$ just to name a few.