## Math 6 Chapter 1 Lesson 15: Factoring a number into prime factors

## 1.Theory

### 1.1. What is prime number analysis?

**Example 1:** Write 300 as a product of many factors greater than 1, with each factor doing the same (if possible)?

For example do the following:

300 = 6.50 = 2 . 3 . 2 . 25 = 2 . 3 . 2 . 5 . 5

300 = 3. 100 = 3. 10 .10 = 3. 2 . 5 . 2 . 5

300 = 3 . 100 = 3. 4 . 25 = 3 . 2 . 2 . 5 . 5

The numbers 2, 3, 5 are prime numbers. We say that 300 has been factored into primes.

\( \Rightarrow \) To factor a natural number greater than 1 into prime factors is to write the number as a product of prime factors.

**Attention:**

a) The prime factorization form of each prime number is to write the number itself.

b) All composite numbers can be factored into primes

### 1.2. How to factor a prime number.

We can also product the number 300 into prime factors along the vertical column

So 300 = 2.2.3.5.5

Simplified by exponentiation, we get: \(300 = {2^2}{.3.5^2}\)

(In the way of factoring a number, we usually write the prime divisors in order from smallest to largest.)

Comment: No matter how we factor a number into prime factors, we end up with the same result.

**Example 2: **Factor the following numbers into prime factors:

a. 120; b. 900 c. 100 000 won

Solution

a. \(120{\rm{ }} = {2^3}.3.5\)

b. \(900 = {2^2}{.3^2}{.5^2}\)

c. \(100{\rm{ }}000 = {10^5} = {2^5}{.5^5}\)

**Example 3:** Factor the following numbers into prime factors and then tell which primes each number is divisible by?

a. 450 b. 2100

Solution

a.\(450 = {2.3^2}{.5^2}\). 450 is divisible by primes 2, 3, 5

b. \(2100 = {2^2}{.3.5^2}.7\). The number 2100 is divisible by the prime numbers 2, 3, 5, 7.

## 2. Illustrated exercise

**Question 1: **Given \(a = {2^2}{.5^2}.13.\) Is each number 4, 25, 13, 20, 8 a divisor of a?

**Solution guide:**

Every \(4 = {2^2},\,\,25 = {5^2},\,13,\,\,20\, = {2^2}.5\) is a divisor of a because they are present in the factors of a. And \(8 = {2^3}\) is not a divisor of a because in the factors of a there is no \({2^3}\).

**Verse 2: **Write down all the divisors of a, b, c knowing that:

a. \(a = 7.11\) b. \(b = {2^4}\) c. \(c = {3^2}.5\)

**Solution guide:**

a. \(a = 7.11\) has the following divisors: 1, 7, 11, 77

b. \(b = {2^4}\) has divisors: 1, 2, 4, 8, 16

c. \(c = {3^2}.5\) has the following divisors: 1, 3, 5, 9, 15, 45

**Question 3:** In a division, the divisor is 86, the remainder is 9. Find the divisor and quotient.

**Solution guide:**

Call divisor b, quotient x, we have:

86 = b . x + 9, where 9 < b

We have: b . x = 86 – 9 = 77.

Infer: b is a divisor of 77 and b > 9. Factorize: 77 = 7 . 11

The divisors of 77 that are greater than 9 are 11 and 77. There are two answers.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given \(a = {2^3}{.3^2}.5.7.\) Is each number 5, 6, 8, 9, 10, 15 a divisor of a?

**Verse 2: **Write down all the divisors of a, b, c knowing that:

a. \(a = 3.7\) b. \(b = {3^3}\) c. \(c = 2^2.5\)

**Question 3:** In a division, the divisor is 127, the remainder is 10. Find the divisor and quotient.

### 3.2. Multiple choice exercises Bài

**Question 1: **Which of the following is a prime number?

A. 149

B. 155

C. 162

D. 175

**Verse 2: **Find 2 consecutive natural numbers knowing that their product is 42?

A. 4.5

B. 5, 6

C. 6, 7

D. 7, 8

**Question 3: **The number of the divisors of 81 is .

A. 2

B. 3

C. 4

D. 5

**Question 4: **Calculate the divisor of the number 126?

A. 10

B. 12

C. 14

D. 16

**Question 5: ** Choose the correct prime factorization

A. 98 = 2.49

B. 145=5.29

C. 81=9.9

D. 100=2.5.10

## 4. Conclusion

Through this lecture, you need to complete some of the objectives given by the lesson, such as:

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