Modulo: Calculate really large numbers without a calculator working on a task: "Recall that $a \equiv b~[n]$ means that there exists an integer $k$ suck that $b = a + k \cdot n$. Are the following claims true or false? 
5.a)  $3 \equiv 5~[10]$
5.b)  $4 \equiv 44~[10]$
5.c)  $298709869876987655 \equiv 809809775~[10]$
I have done 5.a) and 5.b), I think. What I have done is pretty easy: $5 = 3 + 10 \cdot k$, and I set $k$ alone and got $k = (2/10)$, which means the claim is false. 
I did the same for 5.b) which gave $k = 4$, which gives the claim is true.
But this is a bit hard in 5.c) cause of the large number.
Is there a better way to do this or to simplify the task?
And if this is wrong, please show me what I am suppose to do xD 
Thanks! :D
NOTE: I am suppose to do this without a calculater!
 A: For 5c it has already been mentioned in the comments. But let me try to provide a bit of detail.
Two numbers $n$ and $m$ are equal modulo $5$ when $5$ divides their difference. For example (in your notation) $3 \equiv 13\; [10]$ because $10$ divides $13 - 3 = 10$.
Another example if $15 = -25\; [10]$ because $15 - (-25) = 40$ is divisible by $10$.
In general then, if you have a number $n$ that where the last digit is $5$, then $n = 10m +5$ for some integer $m$. Let now $n'$ be another number where the last digit is a $5$. Then there is an integer $m'$ such that $n' = 10m' + 5$. Now look at their difference
$$
n - n' = (10m + 5) = (10m' + 5) = 10(m+m')
$$
This is clearly divisible by $10$ and so $n \equiv n'\; [10]$.
So since both numbers in your question (5c) have last digit $5$, they are equal modulo $5$.
A: Recall that $a \equiv b \pmod n$ if $a$ and $b$ give the same remainder upon dividing by $n$.  For example, if you divide each of $12$ and $77$ by $5$, the remainder is $2$ in both cases. Hence, $12 \equiv 77 \pmod 5$.  Another way to say this is that $a \equiv b \pmod n$ iff $a-b$ is a multiple of $n$.  In the example just given, $12-77$ is a multiple of $5$.
In the special case $n=10$, observe that $a \pmod {10}$ is just the units digit of $a$. For example, if you divide $12768$ by $10$, the remainder is the units digit $8$. Hence, $12768 \equiv 8 \pmod {10}$.  It should now be clear that 5a) is false, 5b) is true and 5c) is true.
