Question

There are two positive integers $X$ & $Y.$ When $X$ is divided by $237,$ the remainder is $192.$ When Y is divided by $117$ the quotient is the same but the remainder is $108.$ Find the remainder when the sum of $X$ & $Y$ is divided by $118.$

Answer 1

Let the quotient obtained in both the cases be ${}^{\prime }q{.}^{\prime }$

We can write $X$ and $Y$ in terms of dividend, remainder, and quotient.

Quotient ${}^{\prime }{q}^{\prime }$ is the same in both cases.

$X=237\times q+192$ [$\because$ Dividend $=$ Divisor$\times$ Quotient $+$ Remainder]

$Y=117\times q+108$

Adding both the equations, we have

$X+Y=(237\times q+192)+(117\times q+108)$

$=(237+117)\times q+192+108$

$=354\times q+300$

$=118\times 3q+118\times 2+64$ [We know $354=3\times 118$ and $300=236+64=2\times 118+64$]

$=118\times (3q+2)+64$ [$\because$ Dividend $=$ Divisor$\times$ Quotient $+$ Remainder]

Thus when the sum of $X$ and $Y$ is divided by $118$ we get $3q+2$ as quotient and $64$ as remainder.

Therefore, the remainder obtained when sum of $X$ & $Y$ is divided by $118$ is $64.$