document.write( "Question 1208954: not sure how to prove this?\r
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\n" ); document.write( "\n" ); document.write( "if n is odd then n^2 = 1 (mod 4)\r
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\n" ); document.write( "\n" ); document.write( "thanks! \n" ); document.write( "

Algebra.Com's Answer #847496 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "n = an odd integer
\n" ); document.write( "n = 2k+1 for some integer k
\n" ); document.write( "n^2 = (2k+1)^2
\n" ); document.write( "n^2 = 4k^2+4k+1 after using the FOIL rule
\n" ); document.write( "n^2 = 4*(k^2+k) + 1
\n" ); document.write( "n^2 = 4*integer + 1
\n" ); document.write( "The last equation shows that whatever n^2 happens to be, it's 1 more than a multiple of 4. This only applies when n is odd.\r
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\n" ); document.write( "\n" ); document.write( "We have shown that (n^2)/4 gives some quotient remainder 1 when n is odd.
\n" ); document.write( "And also proves n^2 = 1 (mod 4) when n is odd.\r
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\n" ); document.write( "\n" ); document.write( "Some examples.
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n	n^2	(n^2)/4	n^2 (mod 4)
1	1	0 remainder 1	1
3	9	2 remainder 1	1
5	25	6 remainder 1	1
7	49	12 remainder 1	1
9	81	20 remainder 1	1
11	121	30 remainder 1	1

\n" ); document.write( "Try some other examples out for yourself.
\n" ); document.write( "When doing modular arithmetic, we ignore the quotient to focus on the remainder only.\r
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\n" ); document.write( "\n" ); document.write( "Extra Credit Question: Try to prove that n^2 = 0 (mod 4) when n is even. A hint is that n = 2k.
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