Question

If a number when divided by 3 leaves remainder 2, then it is not perfect square. Do you agree ? Give reasons.​

Answer 1

Answer:

What reminder of any perfect square is divided by 3?

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It is interesting to see that when a perfect square is divided by 3, the remainder is always 0 or 1.

It is zero when the base number (a in case of a^2) is divisible by 3.

For all other numbers, the square when divided by 3, the remainder is always 1.

When the number is divisible by 3, the remainder is obviously going to be zero.

Let’s say the number is not divisible by 3. Such a number is always in the form of 3a + 1 or 3a - 1, for example 4 = 3*1 + 1 and 5 = 3*2 - 1.

Now, when you square 3a + 1, you get 9a^2 + 6a + 1. Now 9a^2 + 6a = 3*(3a^2+2), therefore that portion is divisible by 3 and what remains (remainder) is 1.

Similarly, when you square 3a - 1, you get 9a^2 - 6a + 1. Now 9a^2 - 6a = 3*(3a^2-2), therefore that portion is divisible by 3 and what remains (remainder) is 1.

Thus, in all situations, we find that the remainder is 1 when the starting number is not divisible by 3.