# Thread: prove that n mod 10 = d

1. ## prove that n mod 10 = d

please can you help me with this problem

2. (a) Note: $10 \equiv 0 \ (\text{mod } 10)$

So we have:
\begin{aligned}n & \equiv d_0 + d_1{\color{red}10} + d_2{\color{red}10}^2 + \cdots + d_{k-1}{\color{red}10}^{k-1} + d_k{\color{red}10}^k \\ & \equiv d_0 + d_1({\color{red}0}) + d_2({\color{red}0})^2 + \cdots + d_{k-1}({\color{red}0})^{k-1} + d_k({\color{red}0})^k \quad (\text{mod } 10) \\ & \ \ \vdots\end{aligned}

Use the same method for (b)

3. thanks for your help

so is that the end of the prove for a?
and for part b, is it only i change 10 to 100?

4. (a) I'm sure you can multiply 0 and d_k.

(b) Yep.