Question

.Which integers leave a remainder of 1 when divided by 2

and also leave a remainder of 1 when divided by 3?

Short answer

answer is missing in file

Step-by-step solution

Step1

The Chinese remainder theorem can be applied to systems with moduli that are not co-prime, but a solution to such a system does not always exist. $\{x=5(mod6)x=3(mod8)\cdot =5(mod6)=3(mod8)$Note that the greatest common divisor of the moduli is 2.

Step2

Suppose X and Y are two simultaneous solutions to a system of congruences. Then we have the following congruences –

$x\equiv a_{1}mod{m}_1,,y\equiv a_{1}mod{m}_1$

\dots

$x\equiv a_{n}mod{m}_n,,,y\equiv a_{n}mod{m}_n$

Combining the two congruences in the rows we have

role="math" localid="1668678763600" $x\equiv ymod{m}_1$

\dots

$x\equiv ymod{m}_n$

Exercise 29 shows that

$x\equiv ymod{m}_1\dots m_n$

Hence the solution is unique