KCET · Maths · Mathematical Reasoning
The greatest value of \(\mathrm{x}\) satisfying \(21 \equiv 385\) \((\bmod x)\) and \(587 \equiv 167(\bmod x)\) is
- A 156
- B 32
- C 28
- D 56
Answer & Solution
Correct Answer
(C) 28
Step-by-step Solution
Detailed explanation
We know that,
\(\quad \mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{x})=\frac{(\mathrm{a}-\mathrm{b})}{\mathrm{x}}\)
Given, \(\quad 21 \equiv 385(\bmod \mathrm{x})=\frac{(21-385)}{\mathrm{x}}\)
\(=-\frac{364}{\mathrm{x}} \quad \text{...(i)}\)
and \(587 \equiv 167(\bmod x)\)
\[
=\frac{(587-167)}{x}=\frac{420}{x} \quad \text{...(ii)}
\]
Now, the greatest value of ' \(x\) ' satisfying Eq. (i) and Eq. (ii) \(=\max [\mathrm{LCM}\) of \((364,420)\) ]
\[
\begin{array}{ll}
\Rightarrow & x=\max (13,15,28) \\
\Rightarrow & x=28
\end{array}
\]
\(\quad \mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{x})=\frac{(\mathrm{a}-\mathrm{b})}{\mathrm{x}}\)
Given, \(\quad 21 \equiv 385(\bmod \mathrm{x})=\frac{(21-385)}{\mathrm{x}}\)
\(=-\frac{364}{\mathrm{x}} \quad \text{...(i)}\)
and \(587 \equiv 167(\bmod x)\)
\[
=\frac{(587-167)}{x}=\frac{420}{x} \quad \text{...(ii)}
\]
Now, the greatest value of ' \(x\) ' satisfying Eq. (i) and Eq. (ii) \(=\max [\mathrm{LCM}\) of \((364,420)\) ]
\[
\begin{array}{ll}
\Rightarrow & x=\max (13,15,28) \\
\Rightarrow & x=28
\end{array}
\]
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