Question
Thu July 05, 2012 By: Dhilip N

show that the cube of any positive integer leaves the remainders 0,1,2 when divided by 3

Expert Reply
Thu July 05, 2012
We know that the cube of any positive integer (say, a) is of the form 9m or 9m+1 or 9m+8 for some integer m.
Case 1: When a = 9m = 3(3m)
It is completely divisible by 3. So, remainder is 0.
 
Case 2: When a = 9m + 1 = 3(3m) + 1
It is not completely divisible by 3. Clearly, in this case when a is divided by 3, then remainder is 1.
 
Case 3: When a = 9m + 8 = 3(3m + 2) + 2
It is not completely divisible by 3. Clearly, in this case when a is divided by 3, then remainder is 2.
Hence, the cube of any positive integer leaves the remainders 0,1,2 when divided by 3.
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