Mathematics Grade 9561.2.9Euclid's Division AlgorithmAThe division algorithmACTIVITY 1.191Is the set of non-negative integers (whole numbers)closed underdivision?2Consider any two non-negative integersaandb.aWhat does the statement "ais a multiple ofb" mean?bIs it always possible to find a non-negative integercsuch thata = bc?Ifaandbare any two non-negative integers, thena÷b(b≠0) is some non-negative integerc(if it exists) such thata = bc.However, since the set of non-negative integers isnot closedunder division, it is clear that exact division is not possible for every pair of non-negativeintegers.For example, it is not possible to compute 17÷5 in the set of non-negative integers, as17÷5 is not a non-negative integer.15 = 3×5 and 20 = 4×5. Since there is no non-negative integer between 3 and 4, andsince 17 lies between 15 and 20, you conclude that there is no non-negative integercsuch that 17 =c×5.You observe, however, that by adding 2 to each side of the equation 15 = 3×5 you canexpress it as 17 = (3×5) + 2. Furthermore, such an equation is useful. For instance itwill provide a correct answer to a problem such as: If 5 girls have Birr 17 to share, howmany Birr will each girl get? Examples of this sort lead to the following theorem calledtheDivision Algorithm.In the theorem,ais called thedividend,qis called thequotient,bis called thedivisor, andris called theremainder.Example 1Writeain the formb×q + rwhere 0≤r < b,aIfa= 47 andb= 7bIfa= 111 andb= 3cIfa= 5 andb= 8Theorem 1.4Division algorithmLetaandbbe two non-negative integers andb≠0, then there existunique non-negative integersqandr,such that,a = (q×b) + rwith0≤r < b.