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Topic: Odd number

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	[No title] (Site not responding. Last check: 2007-11-07)
		The number zero is even, because it is equal to two multiplied by zero.
		A number expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd.
		All prime numbers are odd, with one exception: the prime number 2.
	www.informationgenius.com /encyclopedia/e/ev/even_and_odd_numbers.html (404 words)

	Even and odd numbers : Even number (Site not responding. Last check: 2007-11-07)
		The number zero is considered to be even, because it is equal to two multiplied by zero.
		A number expressed in the decimal number system is even or odd according to whether its last digit is even or odd.
		An integer is odd if it is congruent to 1 modulo this ideal, in other words if it's congruent to 1 modulo 2, and even if it is congruent to 0 modulo 2.
	www.freetemplate.ws /ev/even-number.html (862 words)

	Lesson 8 - Odd and Even Numbers (Site not responding. Last check: 2007-11-07)
		Then, ask for a volunteer to approach the chart and point to the numbers that would be the "A" (the odd numbers) and the numbers that would be the "B." Write the letter A in the odd column and write the letter B in the even column.
		Repeat this exercise with a number in the 20's, the 30's and then with larger numbers that are relevant to the children, such as the number of students in their grade or school.
		For those children having difficulty identifying whether the number is odd or even and/or explaining the rationale, make a note and re-assess them at a later date after continued practice with counting chips and placing them into pairs to determine if the number is odd or even.
	www.michigan.gov /scope/0,1607,7-155-10710_10722_10728-49759--,00.html (1457 words)

	Even and Odd Numbers
		Odd numbers are numbers that cannot be divided evenly by 2.
		An odd number ends in 1, 3, 5, 7, or 9.
		For example, you can't say that the fraction 1/3 is odd because the denominator is an odd number.
	www.factmonster.com /ipka/A0876700.html (245 words)

	Lesson II
		This is another simple lesson, note that there is now one added twist "add 5 if the 'number' is odd." This is another frequently used method in this multiplication system, and it always and only refers to the "number" (see definitions) in its raw state...
		and has nothing to do with the result of doubling or halving the "number." Note that if the Rule were written in the proper order of execution, it would read: if the "number" is odd add 5; either way add half the neighbour.
		Thus far all of the numbers I have used as multiplicands have been long numbers, but you can use the same methods for single digit multiplicands.
	www.cs.unm.edu /~sheppard/math/math-2b.htm (1589 words)

	S(n): A strictly binary tree with n nodes has an odd number of nodes (Site not responding. Last check: 2007-11-07)
		S(n): If n is a number of nodes in a strictly binary tree then n is an odd number.
		Number n is number of nodes in SBT
		Since number of nodes is more than 1 besides the root node it contains two nonempty subtrees L and R. Let R have r nodes.
	members.cox.net /ipevac/cs501CompleteInduction.html (283 words)

	Even or Odd?
		Odd numbers have 1, 3, 5, 7, or 9 in the ones place.) Explain that when skip counting begins with zero, the numbers are even.
		Using an overhead spinner, spin to determine whether their number should be even or odd.
		If the spinner and the number are both even or both odd the student remains in the game.
	www.glc.k12.ga.us /BuilderV03/LPTools/LPShared/lpdisplay.asp?LPID=12243 (696 words)

	PlanetMath: even number (Site not responding. Last check: 2007-11-07)
		The concept of even and odd numbers are most easily understood in the binary base.
		Cross-references: fundamental theorem of arithmetic, induction, odd, states, base, binary, integer
		This is version 5 of even number, born on 2003-09-05, modified 2004-03-06.
	planetmath.org /encyclopedia/EvenNumber.html (86 words)

	Lesson 5 - Odd and Even Numbers (Site not responding. Last check: 2007-11-07)
		They discover several relationships about numbers, including that odd numbers of things always have one left over when objects are grouped in twos, even numbers of objects can be divided equally into two sets, and odd numbers are one more or one less than even numbers.
		Have a child choose a number, hold it up to the class, read it and decide if it is "even" or "odd." If the child has difficulty, refer the class back to the pattern that was discovered on the classroom chart to help.
		The house numbers on one side of the street are "even" and the house numbers on the other side of the street are "odd." Discuss the reason for this pattern with the children.
	www.michigan.gov /scope/0,1607,7-155-10710_10733_10737-49805--,00.html (1291 words)

	Prime and odd numbers (Site not responding. Last check: 2007-11-07)
		3 is prime, 5 is prime, 7 is prime, and by induction we have that all the odd integers are prime.
		There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1).
		3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime,...
	www.gdargaud.net /Humor/OddPrime.html (840 words)

	WKNC (Site not responding. Last check: 2007-11-07)
		The mathematical distinction is that any even number can be written as 2k ie is divisible by two, while a odd* number can be written as 2k+1 and is clearly not divisible by 2.
		As a previous poster mentioned an even number is fundamentally defined as 2k, where k is a member of the integers, and odd numbers are the numbers 2k+1 where k is a member of the integers.
		The idea of even and odd are only a philosophical illusion to the extent that the idea of axiomatic mathematics itself is philosophical illusion, which is perhaps true, since it's not a closed system and can't describe everything.
	wknc.org /forum/read.php?f=1&i=73339&t=72828 (1555 words)

	Sam Loyd's fifteen
		Whenever a counter with a greater number on it precedes a counter with a smaller number, the counters are said to be inverted.
		The reason for the incorporation of the row number in the 4×4 puzzle is that when a counter is moved vertically it jumps over 3 (an odd number of) other counters.
		Thus the difference in the number of inversions before and after the move is always odd.
	www.cut-the-knot.org /pythagoras/fifteen.shtml (843 words)

	Orbix (Site not responding. Last check: 2007-11-07)
		The table shows the number of positions there are for each number of moves.
		In this case the numbers are simply the binomial coefficients, because of the nature of an Abelian group i.e.
		Generally, in any position that is an odd number of moves from being solved, the next button pressed should have an odd number of lit neighbours.
	www.geocities.com /jaapsch/puzzles/orbix.htm (1757 words)

	Even and Odd Numbers
		The concept of even and odd numbers should be revisited daily for the remainder of the year.
		Next, call out a number and have the student partners take turns counting out that number of beans and determine if the number you called out is even or odd.
		Students will put in even number of balloons in the bears paw on the even side and an odd number of balloons on the odd side.
	www.glc.k12.ga.us /BuilderV03/LPTools/LPShared/lpdisplay.asp?LPID=12283 (864 words)

	odd and even idea bank
		The rectangles for the numbers 1-12 have both the number and that amount of small stickers lined up to visually show the one that doesn't have a partner if it is odd.
		They should also be able to discuss that odd numbers end in 1,3,5,7,9 and that two equal groups are not possible; that odd numbers are the numbers that you do not say when counting by 2's.
		Assess their understanding by giving the students numbers and having the student state whether it is odd/even and explain how they determined their choice.
	www.mathcats.com /grownupcats/ideabankoddandeven.html (1358 words)

	Untitled
		I carry out the similar exercise for addition of an even number to an odd number and finally, addition of an odd number to an odd number.
		With the sets of number patterns we can easily help our children understand that multiplication is only a shortcut for representing repeated addition of a single number.
		In the previous article we had seen how 8 x 9 (an even number taken odd number of times) could be represented using the number pattern.
	www.suite101.com /print_article.cfm/14077/73280 (582 words)

	Even and Odd Parity (Site not responding. Last check: 2007-11-07)
		When the frame is received, if the number of ones is not even, an error is reported and the frame declared corrupt.
		If one were odd and the other even, every packet would look corrupt, unless it was of course.
		The problem with parity checking is, if an even number of bits are dropped, say two, then the number of ones remains either even or odd and the error is un-noticed.
	campus.champlain.edu /faculty/rogate/osi/datalink/parity.htm (192 words)

	Perfect Number
		Euclid was one of the first to study perfect numbers, although, due to the difficulty of finding such numbers, he only knew of the first four perfect numbers: 6, 28, 496, and 8128.
		Constructing proofs for when a particular type of odd number cannot be perfect is fine and dandy, but at some point, my method for tackling the odd perfect number problem had to change.
		In his book Perfect Numbers, Richard Shoemaker defines social numbers "as being a chain of numbers in which each number is equal to the sum of all the proper divisors of the preceding number, the last being considered as preceding the first number of the chain" (page 27).
	math.arizona.edu /~rta/001/gaberdiel (6794 words)

	Odd Perfect Numbers (Site not responding. Last check: 2007-11-07)
		In fact, not a single odd perfect number has been found, and whether or not one exists is one of the oldest questions mathmeticians are asking.
		After writing the FSD for each number 1-30, it struck me that the FSDs for odd numbers were all negative, while the FSDs for even numbers were both positive and negative.
		Because, if it could be shown that the FSD for any odd number is negative, then it can be proven that there are no odd perfect numbers (remember: the FSD must equal zero for it to be perfect).
	lucasoman.com /projects/oddperf (1485 words)

	Grade 2: Odd and even numbers
		Ask the students to build the number 7 and the number 10 by using students, where each pair of students holds hands.
		Ask students to indicate whether each number is odd or even and to explain by using a diagram.
		Have children investigate what happens when: (1) two odd numbers are added, (2) two even numbers are added, (3) an odd and even number are added, and (4) an even and odd number are added.
	io.uwinnipeg.ca /~jameis/PAGES/EYR63.html (624 words)

	Is zero considered an even or odd number, or neither? : Zero
		Because you can pick an even number and divide it in groups, take, e.g., 2, which can be divided in two groups of "1", and 4 can be divided in two groups of "2".
		An even number n for which also holds is called a singly even number, while an even number n for which is called a doubly even number.
		A number which is not zero is said to be nonzero.
	www.answerbag.com /q_view.php/1377 (1454 words)

	Odd VIN Number for Tradewind - Airstream Forums
		1959 was the first year California used the "trailer length /bed type /sequential number" format on all their trailers, although some late 1958 models had this VIN format too.
		The 0 before 170 is zero in a set of progressive numbers, not an alphabetical "O" for Ohio and not a zero denoting 1960.
		I suspected I was misreading the numbers and am happy to have this clarified.
	www.airstreamforums.com /forums/showthread.php?t=6084 (580 words)

	Player 1 attempting for a common factor
		Observation 1: If there are an odd number of values to start with, then on Player 1's turn, there will always be an odd number to choose from, and on Player 2's turn there will always be an even number.
		This is because any two numbers that are even, or are odd but divisible by 3 have a common factor.
		Since there are 3 numbers left, but only two categories, there must be a pair that are either congruent mod 2 or congruent mod 3, so Player 1 can simply remove the value that isn't part of the pair to win.
	www.cs.umd.edu /users/gasarch/papers/gameProof/node1.html (1215 words)

	Math 3250/5250 - Solutions 1
		You need to imagine the world consisting of “even-shakers” and “odd-shakers” (those who have shaken an even number of hands and those who have shaken and odd number of hands).
		The total number of handshakes is also the sum of all handshakes by the even-shakers and all handshakes by the odd-shakers.
		Yes: the sum of all handshakes by the odd-shakers can be even only if there is an even number of odd-shakers (the sum of a bunch of odd numbers is even only if there is an even number of numbers).
	www-math.cudenver.edu /~wbriggs/3250f01/3250_sol2.html (689 words)

	c++ - How to read a even or odd from a number? - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-11-07)
		Now the problem is that even if I put in a odd number like 55, the program says 'even', not odd like it should.
		The last bit on all odd numbers (1, 3, 5, 7, 9) is 1.
		For each bit in the original two numbers, the corresponding bit in the third number is a 1 if and ONLY IF the two original bits are both 1.
	www.gamedev.net /community/forums/ViewReply.asp?id=803943 (871 words)

	Benchmark: Math 8.03 (Site not responding. Last check: 2007-11-07)
		An even number is a whole number that is divisible by two (2).
		An odd number is a whole number that is not divisible by two (2).
		If a whole number is not an even number then it is an odd number.
	literacy.owcc.net /abe/Math-B/M-8-03.htm (206 words)

	How to do Odd Count Flat Peyote Stitch (Beading) (Site not responding. Last check: 2007-11-07)
		In Odd Count Flat Peyote, the thread tail is to your left, and the numbering of the beads starts on the left.You then string on another (8th) bead.
		With the number ‘8' it is hard to show it, but this 8th bead is automatically being flipped upside down to land on top of bead #7, when you pull the thread through bead # 6.
		Later, on each odd numbered row, you will do a ‘double loop' to secure the last bead of a row in place.
	members.aol.com /T2945/odpeyins.htm (1466 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		(I believe that the question of whether odd perfect numbers exist is the oldest open question in mathematics, now about 2 and a half millenia old!) It is known that an even number is perfect iff it is of the form 2^(p-1) * ((2^p) - 1) where the second factor is prime.
		As is well known, an odd perfect number $N$, if one exists, must be of the form $N=p\sp \alpha q\sp {2\beta\sb 1}\sb 1 q\sp {2\beta\sb 2}\sb 2\cdots q\sp {2\beta\sb \tau}\sb t$ for distinct odd primes $p,q\sb 1,q\sb 2,\cdots,q\sb t$, with $p\equiv\alpha\equiv1\bmod4$.
		They also show that if $x$ is the number of prime powers $q\sp {2\beta\sb i}\sb i$ for which both $q\sb i\equiv1\bmod 4$ and $\beta\sb i\equiv1\bmod2$, then $p-\alpha\equiv4x\bmod8$.
	www.math.niu.edu /~rusin/known-math/98/perfect (596 words)

	Conjecture 11. Is there any odd Perfect Number ?
		A perfect number is a number equal to the sum of the aliquot part of its proper divisors.
		Euler showed that all the even perfect numbers are of the form given by Euclid.
		Neither we have devised one of these named odd perfect numbers, nor nobody has proved the nonexistence of those odd perfect numbers.
	www.primepuzzles.net /conjectures/conj_011.htm (179 words)

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