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



Answer :

Sure! Let's solve this problem step-by-step using modular arithmetic and the given congruences:

1. Identify the given congruences:
- When the number \( x \) is divided by 9, the remainder is 8.
- When the quotient obtained from dividing \( x \) by 9 is divided by 11, the remainder is 9.
- When the quotient obtained from the previous step is divided by 13, the remainder is 8.

2. Express the problem using congruences:
- \( x \equiv 8 \ (\text{mod} \ 9) \)
- Let \( a \) be the quotient when \( x \) is divided by 9. Thus, \( a \equiv 9 \ (\text{mod} \ 11) \).
- Let \( b \) be the quotient when \( a \) is divided by 11. Thus, \( b \equiv 8 \ (\text{mod} \ 13) \).

3. Express \( x \) in terms of \( a \) and \( b \):
- Starting from \( x \equiv 8 \ (\text{mod} 9) \), write \( x \) as:
[tex]\[ x = 9a + 8 \][/tex]
- Since \( a \equiv 9 \ (\text{mod} \ 11) \), write \( a \) as:
[tex]\[ a = 11b + 9 \][/tex]

4. Substitute \( a \) into the expression for \( x \):
- Substituting \( a \) into \( x = 9a + 8 \):
[tex]\[ x = 9(11b + 9) + 8 \][/tex]
- Simplify inside the parentheses:
[tex]\[ x = 99b + 81 + 8 \][/tex]
- Combine the constants:
[tex]\[ x = 99b + 89 \][/tex]

5. Express \( b \) in terms of the given remainder and modulus 13:
- Since \( b \equiv 8 \ (\text{mod} 13) \), write \( b \) as:
[tex]\[ b = 13c + 8 \][/tex]

6. Substitute \( b \) into the expression for \( x \):
- Substituting \( b \) into \( x = 99b + 89 \):
[tex]\[ x = 99(13c + 8) + 89 \][/tex]
- Simplify inside the parentheses:
[tex]\[ x = 1287c + 792 + 89 \][/tex]
- Combine the constants:
[tex]\[ x = 1287c + 881 \][/tex]

7. Determine the remainder when \( x \) is divided by 1287:
- From the final equation, \( x = 1287c + 881 \), the remainder when \( x \) is divided by 1287 is:
[tex]\[ \boxed{881} \][/tex]

So, the remainder when the given number is divided by 1287 is 881.