Question

(a) 0 (b) 1 (c) 3 (d) can't be determinedA number when divided by 5 leaves a remainder of 4, when the double (i.e., twice) of that number is divided by will be: 5 the remainder

Short answer

Answer: 3

Step-by-step solution

Find the general expression for the given number

Since the given number leaves a remainder of 4 when divided by 5, we can represent it as: n = 5k + 4 where k is an integer.

Find the expression for the double of the given number

To find the double of the given number, we simply multiply the expression we found in step 1 by 2: 2n = 2(5k + 4) 2n = 10k + 8

Determine the remainder when the double of the number is divided by 5

Now let's divide 2n by 5 and see what remainder we get: (10k + 8) divided by 5 Since 10k is a multiple of 5, it's clear that it will leave no remainder when divided by 5. So, we're left with the remainder of 8 when divided by 5. The remainder in this case is 3, so the answer is (c) 3.

Key concepts

Remainders

A remainder is what is left over after division. Imagine dividing a cake among friends and having a piece left that doesn't fit perfectly into the equal slices. This left over piece is the remainder.
In math, the remainder occurs when a number, called the dividend, does not divide evenly by another number, the divisor. For example, if you divide 13 by 5, you can fit 2 complete sets of 5 (which makes 10) into 13, but 3 is left over, hence the remainder is 3.
Understanding remainders is crucial in solving problems related to modular arithmetic and number theory, where remainders form foundational components.

Division

Division is like sharing or grouping numbers and is fundamental in arithmetic operations. It involves dividing a number, the dividend, by another number, the divisor, to get a quotient with or without a remainder.
For instance, when dividing 20 by 4, you get a quotient of 5 with no remainder since 4 fits exactly 5 times into 20. However, division gets interesting when the numbers don't fit perfectly, which introduces remainders.
In our problem, understanding how the division of a number can relate to its multiplied form helps identify patterns and predict outcomes, essential for number theory applications.

Integers

Integers are a set of numbers that include all positive whole numbers, negative whole numbers, and zero. They are a key component in number theory and are used extensively in arithmetic calculations.
Integers can be thought of as the building blocks of standard mathematics, as they help define number properties and operations.
In the exercise, the use of integers, such as the symbol 'k', helps to describe variables and allows the expression of numbers in general forms, critical for solving remainder problems logically and systematically.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations such as addition or multiplication to describe a quantity.
In problem-solving, expressions help represent real-world scenarios using mathematical language. For example, in our exercise, the expression \(n = 5k + 4\) represents any number that, when divided by 5, leaves a remainder of 4.
Writing mathematical expressions clearly is crucial for understanding and solving problems efficiently, allowing insights into number behaviors and aiding in the exploration of abstract mathematical concepts.