What is the remainder when the square of an odd natural number is divided by 8?
| August 8, 2021 | education
An odd natural number can be represented as: 2 * n + 1, where n is some natural number.
(2 * n + 1) ^ 2 = 4n ^ 2 + 4n + 1 = 4n (n + 1) + 1.
The numbers n and (n + 1) are consecutive natural numbers, therefore, among them, one must be divisible by 2.
Therefore, the term 4n (n + 1) must be divisible by 8 without a remainder.
This means that the remainder when dividing the expression (4n (n + 1) + 1) by 8 will be equal to 1.
Answer: The remainder when dividing the square of an odd natural number by 8 is always 1.
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