Question

Show that the square of an odd positive intege can be of the form 6q+1 or 6q+3 for some integer .

Answer 1

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Answer 2

Write the sequence of perfect squares. Let me write first 18 perfect square.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324,...

Divide each term by 6 and write down the sequence of remainders thus formed in order.

1, 4, 3, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 3, 4, 1, 0,...

Here the sequence is being repeated by the sequence 1, 4, 3, 4, 1, 0 over and over.

Find which are the remainders got when odd perfect squares were divided and which are those got when even perfect squares were divided.

When odd perfect squares were divided, the remainders got are 1 and 3. Let me write the sequence of remainders got by dividing odd perfect squares by 6.

1, 3, 1, 3, 1, 3, 1, 3,...

When even perfect squares were divided, the remainders got are 0 and 4. Let me write the sequence of remainders got by dividing even perfect squares by 6.

4, 4, 0, 4, 4, 0, 4, 4, 0,...

As 1 and 3 are got by dividing odd perfect squares by 6, we can write odd perfect squares as either 6q+1 or 6q+3.

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