document.write( "Question 1085899: Prove that one and only one out of n,n+4,n+8,n+12 and n+16 is divisible by 5,where n is any pasitive integer \n" ); document.write( "

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\n" ); document.write( "Without loss of generality, assume n mod 5 = 0 (i.e. assume it is 'n' that 5 divides evenly. It will be clear why this doesn't change anything):
\n" ); document.write( "ThenЕ
\n" ); document.write( "Ч
\n" ); document.write( "n mod 5 = 0
\n" ); document.write( "(n+4) mod 5 = 4
\n" ); document.write( "(n+8) mod 5 = 3
\n" ); document.write( "(n+12) mod 5 = 2
\n" ); document.write( "(n+16) mod 5 = 1
\n" ); document.write( "ЧЧЧЧЧЧ
\n" ); document.write( "if we continued along larger values of nЕ
\n" ); document.write( "(n+20) mod 5 = 0
\n" ); document.write( "(n+24) mod 5 = 4
\n" ); document.write( "(n+28) mod 5 = 3
\n" ); document.write( "(n+32) mod 5 = 2
\n" ); document.write( "(n+36) mod 5 = 1
\n" ); document.write( "ЧЧЧЧЧЧ
\n" ); document.write( "and continuing toward smaller values of nЕ
\n" ); document.write( "(n-4) mod 5 = 1
\n" ); document.write( "(n-8) mod 5 = 2
\n" ); document.write( "(n-12) mod 5 = 3
\n" ); document.write( "(n-16) mod 5 = 4
\n" ); document.write( "(n-20) mod 5 = 0
\n" ); document.write( "Ч
\n" ); document.write( "Now, if it were any other value 5 divided evenly (say n+12 mod 5 = 0) then just let m = n+12 and now the problem hasn't changed: m mod 5 = 0 and m replaces n in our table.
\n" ); document.write( "Ч\r
\n" ); document.write( "\n" ); document.write( "Another, simpler way to look at is is that given n, n+4, n+8, n+12, and n+16, each will give one of the remainders 0,1,2,3,4 when divided by 5. \r
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