Division Principles And Algorithm

Student applies the division algorithm to solve exact divisibility adjustments.

5 lessons

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Concept

Properties of Division

Division by zero, division by one, inverse facts.

When working with whole numbers, division behaves in specific, predictable ways. Let's review the core properties.

Property 1 & 2

Property 1: The quotient of two whole numbers may or may not be a whole number. (e.g., $6 \div 3 = 2$ , but $3 \div 6$ is a fraction, not a whole number).
Property 2: Any non-zero number divided by itself gives a quotient of $1$ (e.g., $5 \div 5 = 1$ ).

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IMAGE

Division Principles and Algorithm Visual

A structural mapping of Dividend, Divisor, Quotient, and Remainder clarifies the formula.

$A bold editorial infographic mapping the Division Algorithm elements. It shows a long division bracket with clearly labeled parts: the inner number as 'Dividend', the outer left number as 'Divisor', the top number as 'Quotient', and the bottom result as 'Remainder'. It uses a clean data-forward design, high-contrast color blocks, elegant typography hierarchy, geometric shapes, professional magazine layout quality, and a 3-4 color maximum.$

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$Dividend = Divisor \times Quotient + Remainder$

Dividend: the number being divided
Divisor: the number we divide by
Quotient: the whole-number answer
Remainder: what is left over

When $10$ is divided by $3$ :

$10 = 3 \times 3 + 1$

So, the quotient is $3$ and the remainder is $1$ .

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Example 10

Finding What to Subtract

Using remainder to find exact divisibility.

Problem

Find the least number that should be subtracted from $1000$ so that $30$ divides the difference exactly.

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reading

Finding What to Add

Using Divisor-Remainder to find exact divisibility.

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Let us solve the problem of finding the least number that should be added to 1000 so that 35 divides the sum exactly.

First, we divide 1000 by 35, which gives a quotient of 28 and a remainder of .

To determine what must be added, we find the difference between the divisor and the remainder. We calculate this difference as $35 - 20 =$ .

Therefore, should be added to 1000 so that the sum 1015 is exactly divisible by 35. This method ensures we are stepping up to the next multiple rather than subtracting down to a previous one.

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Divisibility Adjustment

Worksheet problem to find number to add.

Find the smallest number to add to 2000 so that the result is divisible by 45.

Find the smallest number that should be added to $2000$ so that the result is divisible by $45$ .

In other words, make this number divisible by $45$ :

$2000 + \boxed{\phantom{00}}$

Hint: First divide $2000$ by $45$ and find the remainder.

Step 1: Divide*

Divide 2000 by 45 and find the remainder.

Step 2: Find how much more is needed*

To reach the next multiple of 45, subtract the remainder from 45.

Step 3: Check the new number*

Add your answer to 2000 and check that the result is divisible by 45.

Final Answer*

Write the smallest number that should be added.

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