Question

Find the remainder when the square of any number is divided by \(4 .\) (1) 0 (2) 1 (3) Either (1) or (2) (4) Neither (1) nor (2)

Short answer

Question: When the square of any number is divided by 4, the remainder is always: (1) 0 (2) 1 (3) Either (1) or (2) (4) None of these Answer: (3) Either (1) or (2)

Step-by-step solution

List the possible remainders when a number is divided by 4

When a number is divided by 4, the possible remainders are 0, 1, 2, and 3.

Find the square of the numbers with the above remainders

The next step is to find the square of the numbers with the remainders mentioned in Step 1. Let's denote the remainder as r. If r = 0, the number is a multiple of 4, and its square will also be a multiple of 4. The remainder is 0 in this case. If r = 1, the number can be represented as 4k + 1 (where k is an integer). The square of this number is: (4k + 1)^2 = 16k^2 + 8k + 1 Since 16k^2 + 8k is a multiple of 4, the remainder in this case is 1. If r = 2, the number can be represented as 4k + 2 (where k is an integer). The square of this number is: (4k + 2)^2 = 16k^2 + 16k + 4 = 4(4k^2 + 4k + 1) This shows that the square of a number with a remainder of 2 is a multiple of 4, and thus, the remainder is 0. If r = 3, the number can be represented as 4k + 3 (where k is an integer). The square of this number is: (4k + 3)^2 = 16k^2 + 24k + 9 We can rewrite this expression as 4(4k^2 + 6k + 2) + 1, which implies that the remainder is 1 in this case.

Analyze the remainders obtained

Based on our observation in Step 2, the remainders obtained when the square of any number is divided by 4 are either 0 or 1. Therefore, the correct answer is (3) Either (1) or (2).

Key concepts

Number Theory

Number theory is a fascinating branch of mathematics that deals with the properties and relationships of numbers, especially integers. It includes various areas such as prime numbers, divisibility, and the famous Remainder Theorem, which is relevant to this exercise.
Number theory is essential for understanding the deep structure of integers and how they relate to each other. Among its many applications, it is crucial in cryptography which is used for secure communication.
When looking at the properties of numbers, we often want to find patterns and predict behaviors, such as when squaring a number and dividing by another number. Understanding these principles helps in analyzing and solving problems that involve numbers and their divisors.

Modular Arithmetic

Modular arithmetic is like a clock; no matter how many hours you count, you always loop back to the starting point after a full cycle. When dealing with numbers under modular arithmetic, we work within a set 'modulus', representing numbers in a wrapped-around manner.
In this exercise, we're working with modulo 4. This means we analyze what happens when numbers are divided by 4 and look at the remainder, which is always an integer from the set {0, 1, 2, 3}.

• Square of a number with remainder 0 when divided by 4 results in a remainder of 0.
• Square of a number with remainder 1 results in a remainder of 1.
• Square of a number with remainder 2 results in a remainder of 0.
• Square of a number with remainder 3 results in a remainder of 1.

This pattern stems from how multiplication works in modular arithmetic, showing that despite the size or intricacy of numbers, their squares yield predictable remainders of 0 or 1 when divided by 4.

Integer Division

Integer division is the process of dividing one integer by another and finding both the quotient and the remainder. It forms the basis of problems involving modular arithmetic and finding remainders.
In this exercise, integer division helps determine the result of squaring a number and then dividing it by 4. Through integer division, we typically express this relationship as:

• When r = 0 or r = 2, the remainder after squaring is 0.
• When r = 1 or r = 3, the remainder is 1.

The division, here, tells us about the distribution of numbers around the modulus, impacting how their squares are aligned when divided by 4.
Understanding integer division along with its quotient and remainder components provides insight into larger patterns of number behavior in modular systems. It equips students to predict and solve these types of exercises efficiently.