Question

Let $S=\left\{\right(a,b):a,bϵZ,0\le a,b\le 18\}$. The number of elements $(x,y)$ in $S$ such that $3x+4y+5$ is divided by $19$ is

• A. $38$
• B. $19$
• C. $18$
• D. $1$

Answer 1

The correct option is B $19$

Maximum value of $3x+4y+5$ will occur at $(18,18)$

$\Rightarrow$ Maximum value$=3\times 18+4\times 18+5=131$

And, the minimum value will occur at $0,0$

$\Rightarrow$ Minimum value$=3\times 0+4\times 0+5=5$

Now, the multiples of $19$ that lie between minimum and maximum value of $3x+4y+5$ are $19,38,57,76,95,114$

$Case1-$ When $3x+4y+5=19$

By trial and error the solutions for this case is-

$(2,2)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case is $1$

$Case2-$ When $3x+4y+5=38$

By trial and error the solutions for this case are-

$(11,0);(7,3);(3,6)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case are $3$

$Case3-$ When $3x+4y+5=57$

By trial and error the solutions for this case are-

$(16,1);(12,4);(8,7);(4,10);(0,13)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case are $5$

$Case4-$ When $3x+4y+5=76$

By trial and error the solutions for this case are-

$(17,5);(13,8);(9,11);(5,14);(1,17)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case are $5$

$Case5-$ When $3x+4y+5=95$

By trial and error the solutions for this case are-

$(18,9);(14,12);(10,15);(6,18)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case are $4$

$Case6-$ When $3x+4y+5=114$

By trial and error the solutions for this case are-

$(15,16)$

$\Rightarrow$ The number of elements $(x,y)$ in $S$ for this case are $1$

Therefore, the total number of elements $(x,y)$ in $S$ which are divisible by 19$=1+3+5+5+4+1=19$

Hence the correct answer is option $B$