ques-find the smallest no. which when divided by 28 gives remainder 8,and when divided by32 gives remainder 12.
Hello, sumit2009!
This can be solved with Modulo Arithmetic,
. . but here is an algebraic solution.
Let $\displaystyle N$ = the number.Find the smallest number which when divided by 28 gives remainder 8,
and when divided by 32 gives remainder 12.
Then: .$\displaystyle \begin{array}{cccc}N &=& 28a + 9 & {\color{blue}[1]} \\ N &=& 32b + 12 & {\color{blue}[2]} \end{array}\;\text{ for integers }a,b$
Equate [1] and [2]: .$\displaystyle 28a + 8 \:=\:32b + 12 \quad\Rightarrow\quad 7a \:=\:8b + 1$
. . Then: .$\displaystyle a \:=\:\frac{8b+1}{7} \quad\Rightarrow\quad a \:=\:b + \frac{b+1}{7}$
Since $\displaystyle a$ is an integer, $\displaystyle b+1$ must be a multiple of 7.
. . The first time this happens is: .$\displaystyle b \:=\:6$
Substitute into [2]: .$\displaystyle N \:=\:32(6) + 12 \quad\Rightarrow\quad\boxed{ N \:=\:204}$