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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Wed May 10, 2017

# i wanted to ask that rational numbers are in the form of p/q where q is not equal to zero. but we can write (root 2)/1. in this root 2 is p and 1 is q. q is not equal to zero. so<, it should be a rational number but always read in maths textbooks and vedio lessons of topperlearning that root 2 is irrational. why?

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