document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=4388>Question 4388</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having a lot of trouble with these proofs; I get so far and alskdjfl....\r<BR>\n" );
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document.write( "prove or give a counterexample: if a and b are integers such that 6 divides both a+b and a-b, then 3 divides both a and b.\r<BR>\n" );
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document.write( "Show that the third power of every integer that is not divisible by 7 is of the form 7k+1 or 7k-1 where k is an integer.\r<BR>\n" );
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document.write( "Prove: if n is an integer, then one of the integers n, n+2, and n+4 is divisible by 3. Use it to show that n=3 is the only integer such that n, n+2, and n+4 are primes.\r<BR>\n" );
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document.write( "Prove or give a counterexample: if d=(a,b), then 3d=(3a,3b)\r<BR>\n" );
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document.write( "Prove or give a counterexample: if a and b are integers such that a^2 divides b^3, then a divides b. \r<BR>\n" );
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document.write( "Prove or give a counterexample: if the product of two integers is divisible by 3, then at least one of the integers is divisible by 3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1995 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=1995\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  You did a sloppy job in the posting without separating different questions and<BR>\n" );
document.write( " no marked numbers.\r<BR>\n" );
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document.write( " Be careful about it.\r<BR>\n" );
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document.write( "  (i) if a and b are integers such that 6 divides both a+b and a-b, then 3 divides both a and b. <BR>\n" );
document.write( " (ii) Show that the third power of every integer that is not divisible by 7 is of the form 7k+1 <BR>\n" );
document.write( " or 7k-1 where k is an integer. <BR>\n" );
document.write( " (iii) Prove: if n is an integer, then one of the integers n, n+2, and n+4 is divisible by 3. <BR>\n" );
document.write( " Use it to show that n=3 is the only integer such that n, n+2, and n+4 are primes. <BR>\n" );
document.write( " (iv) Prove or give a counterexample: if d=(a,b), then 3d=(3a,3b) <BR>\n" );
document.write( " (v) Prove or give a counterexample: if a and b are integers such that a^2 divides b^3, <BR>\n" );
document.write( " then a divides b. <BR>\n" );
document.write( " (v) Prove or give a counterexample: if the product of two integers is divisible <BR>\n" );
document.write( " by 3, then at least one of the integers is divisible by 3.<BR>\n" );
document.write( " proof of (i) : Assume a+b = 6 k, a-b = 6j for some integer k,j<BR>\n" );
document.write( "                then 2 a = 6(k+j) and 2 b = 6(k-j) <BR>\n" );
document.write( "                By cancelling 2,we have  a = 3(k+j) and  b = 3(k-j).<BR>\n" );
document.write( "                This shows 3 divides both a and b. <BR>\n" );
document.write( " proof of (ii) : if 7 is not a divisor of n, <BR>\n" );
document.write( "    (7k+1)^3 = 1 mod 7, (7k+2)^3 = 8 mod 7 = 1 mod 7,<BR>\n" );
document.write( "    (7k+3)^3 = 27 mod 7 = 6 mod 7 = -1 mod 7,<BR>\n" );
document.write( "    (7k+4)^3 = 4^3 mod 7 = 16* 4 mod 7 = 2*4 mod 7 = 1 mod 7,<BR>\n" );
document.write( "    (7k+5)^3 = (-2)^3 mod 7 = -8  mod 7 = -1 mod 7 <BR>\n" );
document.write( "    (7k+6)^3 = (-1)^3 mod 7 = -1 mod 7 <BR>\n" );
document.write( " (iii) Consider the three cases of n mod 3 below:<BR>\n" );
document.write( "       when n = 0 mod 3, --> n+2 = 2 mod 3 and n + 4 = 4 mod 3 = 1 mod 3<BR>\n" );
document.write( "       when n = 1 mod 3 -->  n+2 = 0 mod 3, n+4 = 5 mod 3  = 2 mod 3.<BR>\n" );
document.write( "       when n = 2 mod 3 --> n+2 = 4 mod 3 = 1 mod 3, n+4 = 6 mod 3  = 0 mod 3.<BR>\n" );
document.write( "      This shows among n, n+2, and n+4 , only one of them is divisible by 3. <BR>\n" );
document.write( " <BR>\n" );
document.write( "      Note 2 is the smallest prime,ie 1 is not a prime. <BR>\n" );
document.write( "      If n, n+2, and n+4 are primes(n>=2), since one of them is a multiple of 3. <BR>\n" );
document.write( "      We see that 3 is the only prime which is a multiple of 3, <BR>\n" );
document.write( "      But n+2 > 3, as a prime, we see that n+2 cannot be multiples of 3.<BR>\n" );
document.write( "      Similarly n+4 cannot be multiples of 3.<BR>\n" );
document.write( "      Hence, only n must ba multiple of 3 and so n = 3 .<BR>\n" );
document.write( "      Note,n+2 =5, n+4 =7 are also primes.<BR>\n" );
document.write( "      This shows n=3 is the only integer such that n, n+2, and n+4 are primes.      <BR>\n" );
document.write( "  <BR>\n" );
document.write( " (iv) \"if d=(a,b), then 3d=(3a,3b) \"  is  True.<BR>\n" );
document.write( "     Proof: d = (a,b)--> a = d k, b = dj for some integer k,j , &  (k,j) = 1<BR>\n" );
document.write( "      --> 3a = 3dk, 3b=3dj for some integer k,j , &  (k,j) = 1 --> (3a,3b) = 3d <BR>\n" );
document.write( " (v) \"a^2 divides b^3 --> a divides b\" is False<BR>\n" );
document.write( "     For example: Set a= 2^3 = 8, b = 2^2 = 4, we have  a^2 | b^3 (ie 64 =8^2 | 4^3 = 64)<BR>\n" );
document.write( "     But,  a | b ie 8| 4 is false\r<BR>\n" );
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document.write( " (vi) In general, When p is prime then \"p | ab --> p|a or p|  is True.<BR>\n" );
document.write( "      Assume p is prime and p | ab.<BR>\n" );
document.write( "      If p | a ie a = 0 mod p ,we are done.<BR>\n" );
document.write( "      If a is not divisible by p, then (a,p) = 1, there exists integers n, m such that<BR>\n" );
document.write( "      an + pm= 1 (Eucliden Algorithm) , this means an = 1 mod p.<BR>\n" );
document.write( "     Since p |ab --> ab = 0 mod p --> nab = 1b = b mod p --> b is a multiple of p<BR>\n" );
document.write( "     When the prime p = 3, we have 3 | ab  --> 3 | a or 3 |b.\r<BR>\n" );
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document.write( " Try to read carefully about every step and hope that<BR>\n" );
document.write( " don't ask me what mod means.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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