Question:
The product of two numbers 13 and \(x\), when divided by 11 leaves remainder 7. The least positive such integer \(x\) is:
The product of two numbers 13 and \(x\), when divided by 11 leaves remainder 7. The least positive such integer \(x\) is:
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Use modular inverse to solve congruence equations.
Updated On: Jun 15, 2026
- \(8 \)
- \(20 \)
- \(9 \)
- \(18 \)
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The Correct Option is C
Solution and Explanation
Concept:
Convert problem into modular arithmetic.
Step 1: Form equation.
\[ 13x \equiv 7 \pmod{11} \]
Step 2: Reduce modulo.
\[ 13 \equiv 2 \pmod{11} \Rightarrow 2x \equiv 7 \]
Step 3: Find inverse of 2 mod 11.
\[ 2^{-1} = 6 \]
Step 4: Solve equation.
\[ x \equiv 42 \equiv 9 \]
Step 1: Form equation.
\[ 13x \equiv 7 \pmod{11} \]
Step 2: Reduce modulo.
\[ 13 \equiv 2 \pmod{11} \Rightarrow 2x \equiv 7 \]
Step 3: Find inverse of 2 mod 11.
\[ 2^{-1} = 6 \]
Step 4: Solve equation.
\[ x \equiv 42 \equiv 9 \]
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