AP EAMCET · Maths · Basic of Mathematics
The remainder obtained when \((2 m+1)^{2 n}(m, n \in N)\) is divided by 8 is
- A \(1\)
- B \(2\)
- C \(3\)
- D \(4\)
Answer & Solution
Correct Answer
(A) \(1\)
Step-by-step Solution
Detailed explanation
\((2m+1)^2 = 4m^2+4m+1 = 4m(m+1)+1\) \(m(m+1)\) is even, so \(m(m+1)=2k\) for some integer \(k\). \(4m(m+1)+1 = 4(2k)+1 = 8k+1\) \((2m+1)^2 \equiv 1 \pmod 8\) \((2m+1)^{2n} = ((2m+1)^2)^n \equiv (1)^n \pmod 8 \equiv 1 \pmod 8\) Remainder: \(1\)
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