Question

Consider a number $N=21P53Q4$
Number of ordered pairs $(P,q)$ so that the number $B$ is divisible by $9$, is

• A. $11$
• B. $12$
• C. $10$
• D. $8$

Answer 1

The correct option is A $11$

Using the fact that a number is divisible by 9 if the sum of digits is divisible by 9

Sum of the digits $=15+p+q$

So,

$15+p+q=0\left(mod9\right)$

or $p+q=6\left(mod9\right)$

Also $0\le p\le 9$ and $0\le q\le 9$

The only value of $p+q$ are $3\&12$ which satisfies the condition

So, the solution to the problem is the solution of

$p+q=12$

And

$p+q=3$

So, the ordered pairs are

(0,3),(1,2),(2,1),(3,0),(3,9),(4,8),(5,7),(6,6),(7,5),(8,4) and (9,3)

Therefore there are total 11 number of ordered pairs.