Question

# Find the integer a such that , a\equiv 99 (\bmod 41)\ and\ 100\leq a\leq 140

Analyzing functions
Find the integer a such that
$$\displaystyle{a}\equiv{99}{\left({b}\text{mod}{41}\right)}\ {\quad\text{and}\quad}\ {100}\leq{a}\leq{140}$$

$$\displaystyle{a}\equiv{99}{\left({b}\text{mod}{41}\right)}$$ We can find a such that $$\displaystyle{100}\leq{a}\leq{140}$$ by consecutively adding 41 to 99 until we ibtain a vlue between 100 and 140 $$\displaystyle{a}\equiv{99}{\left({b}\text{mod}{41}\right)}$$
$$\displaystyle\equiv{99}+{41}{\left({b}\text{mod}{41}\right)}$$
$$\displaystyle\equiv{140}{\left({b}\text{mod}{41}\right)}$$ Since between 100 and 140 (including): a=140