# Complete Guide For Class 6th Math Chapter 5 – Prime Time

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The** **chapter** Prime Time** introduces students to the fundamental concepts of prime and composite numbers. It explores the idea that a prime number has exactly two distinct factors: 1 and the number itself, while a composite number has more than two factors. The chapter also covers the concept of divisibility, helping students understand how to determine if one number is divisible by another. Additionally, students learn about the importance of prime factorization, which involves breaking down a number into its prime factors, a crucial skill that lays the foundation for more advanced topics in mathematics.

**Prime Time**

**Common Multiples and Common Factors**

Let’s start the chapter of prime time with a fun game (Idli-Vada Game) that you can play with your friends! Imagine you’re sitting in a circle with other children, and the game is all about numbers.

- One child starts by saying “1”.
- The second player says “2”, and so on.
- But when it’s the turn of multiples of 3 (like 3, 6, 9…), the player should say “idli” instead of the number.
- When it’s the turn of multiples of 5 (like 5, 10, 15…), the player should say “vada” instead of the number.
- If a number is a multiple of both 3 and 5 (like 15), the player should say “idli-vada”!

If a player makes a mistake, they are out of the game. The game continues in rounds until only one player remains.

**Key Questions:**

**For which numbers should players say “idli”?**These would be 3, 6, 9, 12, 15, 18, and so on.**For which numbers should players say “vada”?**These would be 5, 10, 15, 20, 25, and so on.**Which is the first number for which players should say “idli-vada”?**The answer is 15, as it is a multiple of both 3 and 5. Such numbers are called**Common Multiples**.

**Jump Jackpot: A Game of Multiples**

Let’s understand the chapter prime time further with a game. In this game, Jumpy and Grumpy are playing a treasure hunt game. Grumpy places a treasure on a number, say 24, and Jumpy has to jump on multiples of a chosen jump size, starting from 0, to reach the treasure.

For example, if Jumpy chooses a jump size of 4, he will jump on 4 → 8 → 12 → 16 → 20 → 24, successfully landing on 24.

**Other successful jump sizes for 24 include** 2, 3, 6, 8, 12, and 24.

But what happens if there are two treasures, placed on different numbers like 14 and 36? Jumpy needs to choose a jump size that lands on both numbers.

**Factors of 14**: 1, 2, 7, 14**Factors of 36**: 1, 2, 3, 4, 6, 9, 12, 18, 36

**Common Factors of 14 and 36** are 1 and 2. So, Jumpy should choose a jump size of 1 or 2 to land on both treasures.

**Common Factors or Divisors** are the numbers that can divide both numbers exactly.

**Perfect Numbers**

As stated in the chapter prime time, a number for which the sum of all its factors is equal to twice the number is called a **Perfect Number**.

The number 28 is an example of a perfect number. Its factors are 1, 2, 4, 7, 14, and 28. Now let’s explore another very important topic of the chapter Prime Time, named perfect numbers.

**Prime Numbers**

**Definition: **Numbers with only two factors are called Prime Numbers.

**Example**:

Guna and Anshu are arranging figs (anjeer) in boxes.

**Guna**wants to put 12 figs in each box.**Anshu**wants to put 7 figs in each box.

Guna can arrange the 12 figs in multiple ways:

- 1 row of 12 figs
- 2 rows of 6 figs
- 3 rows of 4 figs
- 4 rows of 3 figs
- 6 rows of 2 figs

However, Anshu can only arrange the 7 figs in one way:

- 1 row of 7 figs

This difference is because 12 has more than two factors, while 7 has only two factors—1 and 7. Now let’s explore another topic of the chapter prime time named Prime vs. Composite Numbers.

**Prime vs. Composite Numbers**

**Prime Numbers**have only two factors: 1 and the number itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19.**Composite Numbers**have more than two factors. Examples include 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.

**Note:** The number 1 is neither a prime nor a composite number.

**The Sieve of Eratosthenes: Finding Prime Numbers**

Here’s an ancient method to find prime numbers:

**Cross out 1**because it is neither prime nor composite.**Circle 2**, then cross out all multiples of 2.**Circle 3**, then cross out all multiples of 3.**Circle 5**, then cross out all multiples of 5.

Continue this process until all numbers are either circled (primes) or crossed out (composites). This method is called the **Sieve of Eratosthenes**. Now let’s explore the another topic of prime time names twin primes.

**Twin Primes**

According to the chapter Prime Time, twin primes are pairs of prime numbers with a difference of 2. Examples include 3 and 5, 17 and 19.

**Co-prime Numbers: Safe Keeping Treasures**

In the treasure hunt game, Jumpy has to reach two treasures placed on different numbers using the same jump size, but with a new rule: the jump size of 1 is not allowed.

To ensure Jumpy cannot reach both treasures, Grumpy should place them on numbers that have no common factors other than 1, called **Co-prime Numbers**.

For example:

**12 and 26**are not co-prime because they have a common factor of 2.**4 and 9**are co-prime because they have no common factors other than 1.

**Prime Factorisation: Breaking Down Numbers**

As stated in the chapter Prime Time, when checking if two numbers are co-prime, a systematic approach called **Prime Factorisation** is used.

For example:

**Prime Factorisation of 56**: 56 = 2 × 2 × 2 × 7**Prime Factorisation of 63**: 63 = 3 × 3 × 7

These factors are all primes, and the product of these prime factors gives the original number.

**Key Points**:

- Every number greater than 1 has a prime factorization.
- The idea is the same: keep breaking the composite numbers into factors till only primes are left.
- The number 1 does not have any prime factorization. It is not divisible by any prime number.

Here, you see four different ways to get a prime factorization of 36. Observe that in all four cases, we get two 2s and two 3s. Multiply back to see that you get 36 in all four cases.

**Does the order matter?**

When multiplying numbers, we can do so in any order. The end result is the same. That is why, when two 2s and two 3s are multiplied in any order, we get 36.

Thus, the order does not matter. Usually, we write the prime numbers in increasing order. For example, 225 = 3 × 3 × 5 × 5 or 30 = 2 × 3 × 5.

**Prime Factorization of a Product of Two Numbers**

When we perform prime factorization, we start by expressing a number as a product of two factors. For example, 72 can be written as 12 × 6. We then find the prime factorization of each factor:

**12 = 2 × 2 × 3****6 = 2 × 3**

By combining these, the prime factorization of 72 is:

**72 = 2 × 2 × 3 × 2 × 3**- This can also be written as
**2 × 2 × 2 × 3 × 3**.

Always remember to multiply the factors to verify that you get the original number, 72, in this case.

Notice how each prime factor appears a specific number of times in the factorization of 72, compared to how they appear in the factorization of 12 and 6 individually.

**Using Prime Factorization to Check Co-prime Numbers**

Co-prime numbers are pairs of numbers that have no common prime factors. Let’s explore this with an example:

**56 = 2 × 2 × 2 × 7****63 = 3 × 3 × 7**

Since 7 is a common prime factor in both 56 and 63, they are **not co-prime**.

Consider another pair:

**80 = 2 × 2 × 2 × 2 × 5****63 = 3 × 3 × 7**

Here, there are no common prime factors, so **80 and 63 are co-prime**.

**Additional Examples:**

**40 = 2 × 2 × 2 × 5****231 = 3 × 7 × 11**

Since they have no common prime factors, **40 and 231 are co-prime**.

**242 = 2 × 11 × 11****195 = 3 × 5 × 13**

Again, no common prime factors exist, so **242 and 195 are co-prime**.

**Using Prime Factorization to Check Divisibility**

If one number is divisible by another, the prime factorization of the second number will be a part of the prime factorization of the first. Let’s look at an example:

**Example 1: Is 168 divisible by 12?**

**168 = 2 × 2 × 2 × 3 × 7****12 = 2 × 2 × 3**

Since the prime factors of 12 are included in the prime factorization of 168, **168 is divisible by 12**.

**Example 2: Is 75 divisible by 21?**

**75 = 3 × 5 × 5****21 = 3 × 7**

Since 7 is a prime factor of 21 but not of 75, **75 is not divisible by 21**.

**Example 3: Is 42 divisible by 12?**

**42 = 2 × 3 × 7****12 = 2 × 2 × 3**

Since factor 2 appears twice in 12 but only once in 42, **42 is not divisible by 12**.

**Divisibility Tests: Simplifying Factorization for Large Numbers**

As stated in the chapter Prime Time, finding factors of smaller numbers is straightforward, but what about larger numbers?

Let’s take 8560 as an example. Does it have any factors from 2 to 10?

We can easily determine this without long division by applying simple divisibility rules.

**Divisibility by 10**

To check if a number is divisible by 10, observe the last digit:

**Numbers ending in 0 are divisible by 10**.

** Example: **Is 8560 divisible by 10?

Yes, because it ends in 0.

**Divisibility by 5**

Similarly, a number is divisible by 5 if it ends in 0 or 5.

** Example: **Is 8560 divisible by 5?

Yes, because it ends in 0.

**Divisibility by 2**

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

** Example:** Is 8560 divisible by 2?

Yes, because it ends in 0.

**Divisibility by 4**

#### Checking divisibility by 4 involves looking at the last two digits. If they form a number divisible by 4, then the whole number is divisible by 4.

** Example:** Is 8536 divisible by 4?

Yes, because 36 is divisible by 4.

**Divisibility by 8**

#### For divisibility by 8, check the last three digits. If they form a number divisible by 8, then the entire number is divisible by 8.

** Example: **Is 8560 divisible by 8?

No, because 560 is not divisible by 8.

**Special Numbers and Fun with Prime Numbers**

As stated in the chapter Prime Time, special numbers often have unique characteristics.

Let’s explore some examples:

Consider the numbers **9, 16, 25, 43**. Each has something special:

**9**is the only single-digit number.**16**is the only even number and a multiple of 4.**25**is the only multiple of 5.**43**is the only prime number and not a perfect square.

**A Prime Puzzle**

Lastly, try your hand at a prime number puzzle! It’s a fun way to engage with primes and their properties.

**Rules**: Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

**Example**:

75 | |||

42 | |||

102 | |||

170 | 30 | 63 | X |

**Solution**

5 | 5 | 3 | 75 |

2 | 3 | 7 | 42 |

17 | 2 | 3 | 102 |

170 | 30 | 63 | X |

In conclusion, Chapter 5 of CBSE Class 6th Math, “Prime Time,” provides a foundational understanding of prime and composite numbers, along with essential concepts such as divisibility, common factors, and prime factorization. By exploring the distinction between prime and composite numbers, students can build a strong mathematical base. Prime Time is filled with interactive games and activities that make learning these topics engaging and fun. Whether it’s understanding prime factorization or playing with common multiples, the lessons in Prime Time are crucial for grasping higher-level math concepts. Be sure to explore all the resources iPrep offers for Prime Time to master these essential math skills!

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