(Solved): Find the smallest positive integer for which xmod3=2 and xmod4=3 What is the next smallest integer ...

Find the smallest positive integer for which $$x\mathrm{mod}3=2$$ and $$x\mathrm{mod}4=3$$ What is the next smallest integer with this property? [You will have to do some trial and error, but thinking about divisiblity should lead you to some patterns.] The goal of this exercise is to practice finding the inverse modulo $$m$$ of some (relatively prime) integer $$n$$. We will find the inverse of 5 modulo 17 , i.e., an integer $$c$$ such that $$5c?1(\mathrm{mod}17)$$. First we perform the Euclidean algorithm on 5 and 17 : $$\begin{array}{rl}17& =3?+\\ & =2+1\end{array}$$ [Note your answers on the second row should match the ones on the first row.] Thus $$\gcd (5,17)=1$$, i.e., 5 and 17 are relatively prime. Now we run the Euclidean algorithm backwards to write $$1=17s+5t$$ for suitable integers $$s,t$$. $$s=$$ when we look at the equation $$17s+5t?1(\mathrm{mod}17)$$, the multiple of 17 becomes zero and so we get $$5t=1(\mathrm{mod}17)$$. Hence the multiplicative inverse of 5 modulo 17 is