The given statement the values of remainder r ,when a positive integer a is divided by 3,are 0 and 1 only is FALSE.
According to Euclid's division Lemma, a = b q+ r

Here, b = 3

a = 3 q + r, 

Where 0 ≤ r < 3="" and="" r="" is="" an="" />

Therefore , the values of r can be zero ( 0 ) , one ( 1 ) or two ( 2 ).

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Introduction
When a positive integer is divided by 3, there are only two possible remainders: 0 and 1. This phenomenon can be explained through the use of modular arithmetic and the properties of divisibility.

Modular Arithmetic
Modular arithmetic is a branch of mathematics that deals with integers and their remainders when divided by a fixed integer, known as the modulus. In the case of dividing a positive integer by 3, the modulus is 3.

Properties of Divisibility
There are several properties of divisibility that help explain why the values of the remainder when dividing by 3 are limited to 0 and 1 only. These properties include:
- The sum of the digits of a number is congruent to the number itself modulo 3.
- If a number is divisible by 3, then the sum of its digits is also divisible by 3.
- If a number is congruent to 1 modulo 3, then its square is congruent to 1 modulo 3.

Explanation
Using these properties, we can see that:
- Any number whose digits add up to a multiple of 3 will be divisible by 3, and therefore have a remainder of 0 when divided by 3.
- Any number whose digits add up to a number that is not divisible by 3 will have a remainder of 1 when divided by 3.

Therefore, since the sum of the digits of any positive integer is a fixed value, the possible remainders when dividing by 3 are limited to 0 and 1 only.

Conclusion
In conclusion, the values of the remainder when a positive integer is divided by 3 are limited to 0 and 1 only due to the properties of divisibility and modular arithmetic. Understanding these properties can help us better understand the behavior of integers and their remainders when divided by a fixed modulus.