Key Concepts

Habari Mwanafunzi! Let's Unlock the Secrets of Numbers!

Welcome to the exciting world of Factors! Have you ever tried to share sweets with your friends so that everyone gets the same number and there are none left over? If you have 10 sweets and 5 friends, each person gets 2. Perfect! But what if you have 10 sweets and 3 friends? Hmm, that gets complicated! The magic you used in the first case is all about factors. By the end of this lesson, you'll be a master at dividing and grouping numbers, a skill that is super useful in everything from shopping at the market to planning events. Tusome pamoja! (Let's learn together!)

1. What Exactly is a Factor?

A factor is a number that divides into another number exactly, with no remainder. Think of them as the building blocks of a number.

Imagine Mama Mboga has 12 passion fruits. She wants to arrange them in neat, equal rows on her stall. She could have:

• 1 row of 12 fruits
• 2 rows of 6 fruits
• 3 rows of 4 fruits
• 4 rows of 3 fruits
• 6 rows of 2 fruits
• 12 rows of 1 fruit

The numbers 1, 2, 3, 4, 6, and 12 are the factors of 12 because they allow her to arrange the fruits perfectly!

To find the factors of a number, you just find all the pairs of numbers that multiply to give you that number. Let's find the factors of 18:

1 x 18 = 18
2 x 9  = 18
3 x 6  = 18

So, the factors of 18 are 1, 2, 3, 6, 9, and 18. Easy, right?

Image Suggestion: A vibrant, colourful digital illustration of a Kenyan market stall ('kibanda'). A friendly Mama Mboga is arranging 12 passion fruits into a perfect rectangle (3 rows of 4). The rows are clearly visible. The style is cheerful and educational.

2. The Number Superstars: Prime vs. Composite

Numbers can be sorted into special groups. Two of the most important groups are Prime and Composite numbers.

• A Prime Number is a special number that has only two factors: 1 and itself. They are the unique superstars! Examples: 2, 3, 5, 7, 11, 13...
• A Composite Number is a number that has more than two factors. They are the social butterflies with lots of connections! Examples: 4, 6, 8, 9, 10, 12...

Important Note: The number 1 is a special case. It is neither prime nor composite!

3. Prime Factors: The DNA of a Number

Every composite number can be broken down into a string of prime numbers that multiply together. This is called prime factorization. It's like finding the unique DNA code for that number! The easiest way to do this is by using a "Factor Tree".

Let's find the prime factors of 48 using a factor tree:

        48
       /  \
      2    24      (48 is 2 x 24. 2 is prime, so we circle it!)
           /  \
          2    12    (24 is 2 x 12. 2 is prime!)
               /  \
              2    6     (12 is 2 x 6. 2 is prime!)
                   / \
                  2   3  (6 is 2 x 3. Both are prime!)

Once you have only prime numbers at the "bottom" of your tree's branches, you are done! The prime factors of 48 are all the circled numbers.

So, the prime factorization of 48 is: 2 x 2 x 2 x 2 x 3

4. The Team Captains: HCF and LCM

Now we use our knowledge for something very powerful! Finding what numbers have in common.

Highest Common Factor (HCF)

The HCF (also called Greatest Common Divisor or GCD) is the largest factor that two or more numbers share.

Real-World Problem: A youth group leader has 24 sodas and 36 mandazis for a party. She wants to create identical snack packs to give out, with no items left over. What is the largest number of snack packs she can make?
To solve this, we need to find the HCF of 24 and 36!

1. Step 1: Find the prime factors of each number.

24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
    

2. Step 2: Identify the common prime factors.

Both numbers share two 2s and one 3.

Common Factors: 2, 2, 3
    

3. Step 3: Multiply the common factors together.

HCF = 2 x 2 x 3 = 12
    

Answer: The leader can make a maximum of 12 identical snack packs! (Each pack would have 2 sodas and 3 mandazis). Unaona? Maths in action!

Least Common Multiple (LCM)

The LCM is the smallest number that is a multiple of two or more numbers.

Image Suggestion: A bright, sunny scene at a busy but orderly Kenyan bus stage. Two colourful matatus, one labelled "Route A (Every 15 mins)" and the other "Route B (Every 20 mins)", are shown leaving the station at the same time. The clock above shows 8:00 AM.

Real-World Problem: Two matatus leave the same stage at 8:00 AM. Matatu A returns to the stage every 15 minutes. Matatu B returns every 20 minutes. At what time will they next be at the stage together?
To solve this, we need to find the LCM of 15 and 20!

1. Step 1: Find the prime factors of each number.

15 = 3 x 5
20 = 2 x 2 x 5
    

2. Step 2: List all prime factors, using the highest power of each.

We have 2s, 3s, and 5s. The most 2s we see in one number is two (from 20). The most 3s is one (from 15). The most 5s is one (from either).

Factors to use: 2, 2, 3, 5
    

3. Step 3: Multiply these factors together.

LCM = 2 x 2 x 3 x 5 = 60
    

Answer: The matatus will be at the stage together after 60 minutes, which is 1 hour. So, they will meet again at 9:00 AM!

You've Got This!

Hongera! You have just learned the key concepts of factors. You can now see numbers not just as single items, but as complex things with their own building blocks (factors) and family trees (multiples). This is a foundation for so much more in mathematics. Keep practicing, stay curious, and you will excel!