Lesson Product of two consecutive integers is divisible by 2
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<H2>Product of two consecutive integers is divisible by 2</H2> It is useful to know that <B>the product of any two consecutive integers is divisible by 2</B>. Indeed, one of the two consecutive integers is even and, hense, is divisible by 2. Therefore, the product of the two consecutive integers is divisible by 2. <H3>Example 1</H3>Without calculations, you can state that the product 1277*1278 is divisible by 2. Below you will find typical word problems dealing with the product of two consecutive integers. <H3>Problem 1</H3>Three persons sent Cristmas postcards each to the other. In total, how many Cristmas postcards were sent? <B>Solution</B> Each of three persons sent 2 Cristmas postcards to the two others. Hence, 3*2 = 6 postcards were sent in total. <B>Answer</B>. In total, 6 Cristmas postcards were sent. <H3>Problem 2</H3>Four persons sent Cristmas postcards each to the other. In total, how many Cristmas postcards were sent? <B>Solution</B> Each of four persons sent 3 Cristmas postcards to the three others. Hence, 4*3 = 12 Cristmas postcards were sent in total. <B>Answer</B>. In total, 12 Cristmas postcards were sent. <H3>Problem 3</H3><B>n</B> persons sent Cristmas postcards each to the other. In total, how many postcards were sent? <B>Solution</B> Each of <B>n</B> persons sent <B>n-1</B> postcards to the others. Hence, <B>n</B>*(<B>n</B>-1) Cristmas postcards were sent in total. <B>Answer</B>. In total, <B>n</B>*(<B>n</B>-1) postcards were sent. <H3>Problem 4</H3><B>n</B> persons sent Emails each to the other. In total, 20 Emails were sent. Find the number of persons exchanging by Emails. <B>Solution</B> Let <B>n</B> be an unknown number of persons exchanging by Emails. Each of <B>n</B> persons sent <B>n-1</B> Emails to the others. Hence, <B>n</B>*(<B>n</B>-1) Emails were sent in total. Thus you have the equation {{{n*(n-1)}}} = {{{20}}} to find the unknown <B>n</B>. Solve it step by step by reducing to the quadratic equation and applying the quadratic formula: {{{n^2}}} - {{{n}}} = {{{20}}}, {{{n^2}}} - {{{n}}} - {{{20}}} = {{{0}}}, {{{n}}} = {{{(1 +- sqrt(1 + 4*20))/2}}} = {{{(1 +- sqrt(81))/2}}} = {{{(1 +- 9)/2}}}. So, you have {{{n[1]}}} = {{{(1 + 9)/2}}} = {{{5}}} and {{{n[2]}}} = {{{(1 - 9)/2}}} = {{{-4}}}. Only positive number might be the solution. Hence, {{{n}}} = {{{5}}} is the answer. <B>Answer</B>. 5 persons were in the group exchanging by Emails. <H3>Problem 5</H3>Three points are given at a plane not lying in one straight line. Every two of them are connected by a straight segment. In total, how many segments are there in the plane connecting the given points? <B>Solution</B> Two segments are released from each of the three points connecting this point with the two others. So, at first glance, there are 3*2 = 6 segments in total. But counting in this way, we count each segment twice. Thus the actual number of the segments is half of 6, i.e. {{{6/2}}} = 3. <B>Answer</B>. In total, there are 3 segments connecting given points. <H3>Problem 6</H3>Four friends played chess games each with the other. In total, how many chess games were played? <B>Solution</B> Each of four persons played 3 chess games with the three others. So, at first glance, there were 4*3 = 12 chess games played in total. But counting in this way, we count each game twice. Thus the actual number of the played games is half of 12, i.e. {{{12/2}}} = 6. <B>Answer</B>. In total, 6 chess games were played. <H3>Problem 7</H3>Jonny marked <B>n</B> points in a plane in a way that no three of them are lying in one straight line. Then Jonny connected every two points by a straight segment. In total, how many segments did Jonny draw in the plane connecting the marked points? <B>Solution</B> <B>n</B>-1 segments are released from each of <B>n</B> points connecting this point with <B>n</B>-1 others. So, at first glance, there are <B>n</B>*(<B>n</B>-1) segments in total. But counting in this way, we count each segment twice. Thus the actual number of the segments is half of <B>n</B>*(<B>n</B>-1), i.e. {{{(n*(n-1))/2}}}. <B>Answer</B>. In total, there are {{{(n*(n-1))/2}}} segments connecting <B>n</B> points. <H3>Problem 8</H3>Jonny marked <B>n</B> points in a plane in a way that no three of them are lying in one straight line. Then Jonny connected every two points by a straight segment. How many points did Jonny marked in the plane if he drew 15 straight segments connecting the marked points? <B>Solution</B> If <B>n</B> is an unknown number of marked points in the plane, then the number of segments connecting these points is {{{(n*(n-1))/2}}}, according to the solution of the previous <B>Problem 7</B>. Thus you have the equation {{{(n*(n-1))/2}}} = {{{15}}} to find the unknown <B>n</B>. Solve it step by step by reducing to the quadratic equation and applying the quadratic formula: {{{n^2}}} - {{{n}}} = {{{30}}}, {{{n^2}}} - {{{n}}} - {{{30}}} = {{{0}}}, {{{n}}} = {{{(1 +- sqrt(1 + 4*30))/2}}} = {{{(1 +- sqrt(121))/2}}} = {{{(1 +- 11)/2}}}. So, you have {{{n[1]}}} = {{{(1 + 11)/2}}} = {{{6}}} and {{{n[2]}}} = {{{(1 - 11)/2}}} = {{{-5}}}. Only positive number might be the solution. Hence, {{{n}}} = {{{6}}} is the answer. <B>Answer</B>. 6 points were marked in the plane.
