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Sagot :
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.
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.
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