Key Concepts

Habari Mwanafunzi! Let's Talk About Sharing!

Imagine you have a packet of 12 delicious Patco sweets. You want to share them with your friends. If you have 2 friends, you can each get 4 sweets, and none are left over. Perfect! But what if you have 4 friends? You'd get 2 sweets each, with 2 left over for the teacher (me!).

This simple idea of sharing things perfectly, with no remainder, is the heart of divisibility. It’s a super-power in mathematics that helps us solve problems much faster. Today, we will master the key ideas behind it. Let's begin!

Image Suggestion: A vibrant, cheerful digital illustration of three Kenyan school children in uniform, happily sharing a plate of golden-brown mandazis under an acacia tree. One child is passing a mandazi to another. The style should be colourful and inviting, like a modern storybook illustration.

What Does "Divisible" Actually Mean?

In mathematics, we say a number is divisible by another number if it can be divided completely, with a remainder of zero. It's a clean, perfect split!

Let's look at 20 shillings. Is it divisible by 5? Yes! Because you can get exactly four 5-shilling coins from it. Is it divisible by 6? No, because you'd have 2 shillings left over.

The Math Breakdown:

20 ÷ 5 = 4
The remainder is 0.
So, 20 IS divisible by 5.

20 ÷ 6 = 3 with a remainder of 2
The remainder is NOT 0.
So, 20 IS NOT divisible by 6.

To understand this better, let's learn the official names for each part of a division problem:

      QUOTIENT (The Answer)
      ___________
DIVISOR | DIVIDEND

Example: 20 ÷ 5 = 4

Dividend: 20 (the number being divided)
Divisor: 5 (the number you are dividing by)
Quotient: 4 (the result)

For a number to be divisible, the math must work out perfectly, leaving no remainder!

Key Concept 1: Factors - The Building Blocks

Factors are like the ingredients of a number. They are all the numbers that can be multiplied together to get that number. Or, to use our new skill, factors are all the numbers that can divide another number perfectly (with a remainder of 0).

Real-World Example: Imagine you are arranging chairs for a harambee (fundraiser). You have 18 chairs. You could arrange them in:

• 1 row of 18 chairs.
• 2 rows of 9 chairs.
• 3 rows of 6 chairs.

The numbers 1, 2, 3, 6, 9, and 18 are all factors of 18 because they allow you to group the chairs perfectly with none left out!

Let's find all the factors of 12, step-by-step:

Always start with 1.
1 x 12 = 12  (So, 1 and 12 are factors)

Try 2. Is 12 divisible by 2? Yes.
2 x 6 = 12   (So, 2 and 6 are factors)

Try 3. Is 12 divisible by 3? Yes.
3 x 4 = 12   (So, 3 and 4 are factors)

Try 4. We already have it! (from 3 x 4)

So, the factors of 12 are: 1, 2, 3, 4, 6, 12.

Key Concept 2: Multiples - The Growing Family

If factors are the building blocks, multiples are the big structures you build! A multiple of a number is what you get when you multiply that number by any whole number (like 1, 2, 3, 4, ...). Think of it as its "times table".

Real-World Example: A matatu has 14 seats for passengers. After one full trip, it has carried 14 people. After two full trips, it has carried 28 people. After three full trips, 42 people. The numbers 14, 28, 42, 56, and so on, are all multiples of 14.

Finding the first few multiples of a number is easy. Let's find the first 5 multiples of 7:

7 x 1 = 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
7 x 5 = 35

The first 5 multiples of 7 are: 7, 14, 21, 28, 35.

The Big Connection: Factors and Multiples are a Team!

Factors and multiples are two sides of the same coin. They have a special relationship. Look at this:

• 3 is a factor of 18.
• This means that 18 is a multiple of 3.

See? It works both ways! If one number divides another perfectly, the smaller one is the factor, and the bigger one is the multiple.

+------------------------------------------+
|                                          |
|  Because 4 x 5 = 20 ...                  |
|                                          |
|  +--> 4 is a FACTOR of 20                |
|  +--> 5 is a FACTOR of 20                |
|                                          |
|  +--> 20 is a MULTIPLE of 4              |
|  +--> 20 is a MULTIPLE of 5              |
|                                          |
+------------------------------------------+

Why is This Important?

Understanding divisibility, factors, and multiples is not just for exams! It helps you:

• Share things fairly without leftovers.
• Solve problems involving patterns and groups.
• Understand more complex topics later on, like fractions and prime numbers.

Key Takeaways

• Divisible: A number can be divided by another number with a remainder of ZERO.
• Factor: A number that divides another number perfectly. (The "ingredient").
• Multiple: The result of multiplying a number by a whole number. (The "final product").
• Factors and Multiples are opposites but are always linked together!

Excellent work today! Keep practicing, and you'll become a divisibility champion in no time. Kazi nzuri!