KCET · Maths · Mathematical Reasoning
Which of the following is incorrect?
If \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\) and \(\mathrm{x}\) is an integer, then
- A \((a+x) \equiv(b+x)(\bmod m)\)
- B \((a-x) \equiv(b-x)(\bmod m)\)
- C \(\mathbf{a x} \equiv \mathrm{bx}(\bmod \mathrm{m})\)
- D \((a+x) \equiv(b \div x)(\bmod m)\)
Answer & Solution
Correct Answer
(D) \((a+x) \equiv(b \div x)(\bmod m)\)
Step-by-step Solution
Detailed explanation
(i) \(a \equiv b(\bmod m)\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}+\mathrm{x})-(b+\mathrm{x})\)
\(\Rightarrow \quad \mathrm{a}+\mathrm{x} \equiv \mathrm{b}+\mathrm{x}(\bmod \mathrm{m})\)
(ii) \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b})\)
\(\Rightarrow \quad m \mid\{(a-x)-(b-x)\}\)
\(\Rightarrow \quad \mathrm{a}-\mathrm{x} \equiv \mathrm{b}-\mathrm{x}(\bmod \mathrm{m})\)
(iii) \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b}) \mathrm{x}\)
\(\Rightarrow \quad a x \equiv b x(\bmod m)\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}+\mathrm{x})-(b+\mathrm{x})\)
\(\Rightarrow \quad \mathrm{a}+\mathrm{x} \equiv \mathrm{b}+\mathrm{x}(\bmod \mathrm{m})\)
(ii) \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b})\)
\(\Rightarrow \quad m \mid\{(a-x)-(b-x)\}\)
\(\Rightarrow \quad \mathrm{a}-\mathrm{x} \equiv \mathrm{b}-\mathrm{x}(\bmod \mathrm{m})\)
(iii) \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\)
\(\Rightarrow \quad \mathrm{m} \mid(\mathrm{a}-\mathrm{b}) \mathrm{x}\)
\(\Rightarrow \quad a x \equiv b x(\bmod m)\)
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