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Asked by Nidhi Munshi | 18 Apr, 2013, 05:14: PM
answered-by-expert Expert Answer
As per the Euclid's division algorithm, any number can be expressed in the following way
 
a = b(q) + r, such that 0<=r<b and q>=0
 
Now, in this case, where we had to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8, we need to express a number in the Euclid's form and then calculate its cube. 
 
Just by looking at the form of the answer we want, it seems like 9 is common across all the 3 cases. Also, there are 3 cases, which means that all numbers in the number series, can be expressed in one of the 3 ways. 
 
Considering these 2 things, it seems logical to consider b as 3, as then we will have 3 ways (with r = 0 or 1 or 2) to represent all the integers in the number series. If we would have considered b as 9, then, r could have 8 different possibilties and hence, we would have initially got 8 different cases, making it very complex. 
Answered by | 18 Apr, 2013, 09:55: PM

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