How to linearize a product of an integer and a binary variable

i have this constraint right here, which is not linear. How would i linearize such a product. $$number_t$$ is a positive integer and $$new_t$$ and $$reset_t$$ are binary.

$$number_t = (number_{t-1}+new_t)\cdot (1-reset_t)$$

\begin{align} 0 \leq number_t \leq (number_{t−1}+new_t) \newline number_t \geq (number_{t−1}+new_t)-reset_tM \newline number_t \leq M(1−reset_t) \end{align}
If $$reset_t = 0$$, through the first and second constraint we force $$number_t$$ to take value $$number_{t-1}+new_t$$. If $$reset_t=1$$, through the first and third constraint, we force $$number_t$$ to be 0. Where $$M$$ is a large enough number.
• Thank you @PeterD. Assuming i also have $number_t = (number_{t-1}+new_t)\cdot (1-reset_t)-c_t + b_t$, which of both are also binary, how would i need to modify the constraints then? Thanks in advance? Feb 13 at 14:38
• Would it work if, for example, I replace $number_t$ with $nr_t$ in your solution and then calculate $number_t=nr_t-c_t+b_t$? Feb 13 at 15:17