Question

On dividing a number by 9, the remainder is 8. The quotient so obtained when divided by 11, leaves the remainder 9. Now the quotient so obtained when divided by 13, leaves the remainder 8. Find the remainder when the given number is divided by 1287.

Answer 1

Remainder Theorem

Given integers $a$ and $b$ with $b>0$, there exist unique integers $q$ and $r$ such that $a=bq+r$, where $0\leqslant r<b$, where $q$ is called the quotient and $r$ is called the remainder in the division of $a$ by $b$.

Solution:

Step 1.

Let $x_{1}$ be the given number, which on division by $9$ gives remainder $8$.

If $x_{2}$ be the quotient, then

$\quad x_{1}=9x_{2}+8$ .....(1)

Step 2.

Here $x_{2}$ is the number, which on division by $11$ gives remainder $9$.

If $x_{3}$ be the quotient, then

$\quad x_{2}=11x_{3}+9$ .....(2)

Step 3.

Here $x_{3}$ is the number, which on division by $13$ gives remainder $8$.

If $q$ be the quotient, then

$\quad x_{3}=13q+8$ .....(3)

Step 4.

Substituting the value of $x_{3}$ in (2), we get

$\quad x_{2}=11(13q+8)+9$

$\Rightarrow x_{2}=143q+88+9$

$\Rightarrow x_{2}=143q+97$ .....(4)

Step 5.

Substituting the value of $x_{2}$ in (1), we get

$\quad x_{1}=9(143q+97)+8$

$\Rightarrow x_{1}=1287q+873+8$

$\Rightarrow x_{1}=1287q+881$ .....(5)

Step 6.

From equation (5), we can conclude that on division by $1287$, the given number $x_{1}$ (as assumed) leaves the remainder $881$.

Answer.

• The required remainder is $\bold{881}$.

Answer 2

Step-by-step explanation:

For some integer n, the last number divided is 13n+8. The dividend before that is

11(13n+8)+9, and the dividend before that is 9(11(13n+8) +9) +8 = 1287n + 881.

given number is divided by 1287 is 881