 let's say we have five flags and during Indian independence day we have to give two flags to one student okay so let's say if I give two flags to one student so how many students can I allot these flags to so two flags to one then these two to another and one is remaining so I cannot give this flag alone or I cannot give a single flag to one student so hence this will be remainder so if you see if we divide the set of five by two so I can I can take out one group of two there and I can take out another group of two but then one this flag will be left out so hence I cannot complete the set of two here so hence we say that if we divide five by two I get two sets of two each and one is remainder this is what Euclid's division lemma explains that if we have if we had five if we had five and we have to give two to one one student each then I can extract out two groups no more one will be left out if I had another flag here then the two set of two would have been completed and then I would have given it to three different students so hence if I divide six by two I would get three isn't it but if I would have divided five by two then I would have got one plus one two so that becomes my quotient and this becomes my remainder that's what Euclid division lemma explains in this example if you see we have nine pens here and let's say we have to give two pens two pens to one student okay so how many groups of two can I form I can form this one then this another one so two groups then third group then fourth group but I can't complete this group this group so hence if we what does it mean with respect to Euclid's division lemma that we can divide nine so there were nine I could divide nine into let's say set of twos so how many set of twos will I get I'll get one two three and four four set of twos I'll get and one will be remainder so if you see this one the remainder one is less than the divisor which was two which was two and the quotient is nothing but one two three four so hence this one is less than two that's what Euclid's division lemma also says that the remainder must be greater than equal to zero or less than equal to the divisor which is in this case is two now imagine that I have to divide this in the groups of three so one two and three so I could get one two three groups of three so in terms of Euclid's division lemma I can say that while if I divide nine by three I'll get three as quotient and there is nothing here as the remainder so this is another example another example would be let's say I divide this group of pen in the set of four so how many set of fours can I get so that means I am dividing nine by four so one two so two set two sets of four is achieved and one is remainder so hence if you divide nine by four two is the quotient and one is the remainder similarly let's say again if I have to divide nine by five so I can I can create five a set of five a set of five one set of five that means if I divide five nine by five I'll get one as the quotient so one set I could form but these four are remaining now if you notice what was my quotient divisor here five right and the remainder is one two three four which is less than less than five now if if I would have another pen I would have had another pen then I would have completed the set of five once again and hence the quotient would have been two and hence the remainder will automatically drop to zero correct so hence in no none of the cases there will be remainder which will be more than the divisor because the moment it is more than the divisor then you can complete one full set again and that will be counted in the quotient the quotient will be added by one right and hence remainder will always be either so what will be the remainder if any number is divided by let's say five so I will be having either zero as the remainder or I'll get one is one as the remainder when will that be let's say there were six pens and now I have to divide in the in in let's say divide by five so I'll get only one as the quotient and one will be remainder if I would have had seven pens then again one will be the quotient two will be the remainder if I would have had eight pens then one set of five and three is the remainder I would have had nine then five one set of five and four as the remainder right but the moment I would have had the tenth pen then I can complete one set of five again and the quotient gets incremented by one and remainder again becomes zero so hence remainder can go from the value of zero to one to two three to four that means the remainder values could be anything less than all the integers from zero to four that is what was explained by Euclid in his division lemma