## The Fundamental Theorem of Arithmetic

Online shopping has become popular with the younger generation because making digital payments have become safe and easy. Do you know that prime numbers have a huge role to play in making online payments safe? In fact most of cyber-security measures including data encryption rely on properties of prime numbers. One of those properties is that while it is relatively easy to find larger prime numbers, it’s unavoidably hard to factor large numbers back into primes. The RSA algorithm, the most common type of encryption used today, was named after the three mathematicians who first publicly unveiled it in 1977.

In this lesson, we will learn about prime factorisation. And, who knows one day you will discover a much safer way to protect digital economics.

## FACTS USED IN THIS LESSON

- HCF(a, b) = Product of smallest power of each common prime factor of a and b.
- LCM(a, b) = Product of greatest power of each prime factor of a and b.
- HCF(a, b) x LCM(a, b) = a x b
- A number is
**prime**if it has only two divisors, 1 and itself. - A number is
**composite**if it has more than 2 divisors. **Fundamental Theorem of Arithmetic**: Every composite number can be uniquely expressed as a product of two or more prime numbers.

**EXERCISE**: Express each number as product of its prime factors:

i) 140

ii) 156

iii) 3825

iv) 5005

v) 7429

**SELF PRACTICE**: Express each number as product of its prime factors:

i) 260

ii) 312

iii) 13500

iv) 7007

v) 3375

**EXERCISE**: Find the LCM and HCF of 510 and 92. Verify that LCM × HCF = product of the two numbers.

**SOLUTION** The prime factorization of the given numbers is:

510 = 2 × 3 × 5 × 17,

92 = 2 × 2 × 23 = 2^{2} × 23.

HCF (510, 92) = Product of smallest power of each common prime factor of 510 and 92.

So, HCF (510, 92) = 2.

LCM (510, 92) = Product of greatest power of each prime factor of 510 and 92.

So, LCM (510, 92) = 2^{2} × 3 × 5 × 17 × 23 = 23460.

We need to verify that:

The product of given numbers = The product of HCF and LCM.

510 × 92 = = 2 × 23460

46920 = 46920

Hence verified.

**SELF PRACTICE**: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

i. 26 and 91

ii. 336 and 54

iii. 90 and 144

iv. 624 and 72

v. 1008 and 108

**EXERCISE**: Find the LCM and HCF of 12, 15, and 21 by applying the prime factorization method.

**SOLUTION**: The prime factorisation of the given numbers are

12 = 2 × 2 × 3 = 2^{2} × 3,

15 = 3 × 5,

21 = 3 × 7.

The HCF of three number = Products of smallest power of each common prime factor of the numbers.

So, HCF of 12, 15 and 21 = 3.

The LCM of three numbers = Product of greatest power of each prime factor of the numbers.

So, LCM of 12, 15 and 21 = 2^{2} × 3 × 5 × 7 = 420.

**SELF PRACTICE**: Find the LCM and HCF of the following integers by applying the prime factorization method.

i) 17, 23, 29

ii) 8, 9, 25

iii) 24, 30 and 42

iv) 11, 17 and 19

v) 10, 11 and 30

**EXERCISE**: Given that HCF (306, 657) = 9, find LCM (306, 657).

**SELF PRACTICE**: Find the LCM of the pair of numbers whose HCF is given:

i) HCF (128, 184) = 8

ii) HCF (192, 256) = 64

iii) HCF (486, 540) = 54

**EXERCISE**: Check whether 6* ^{n}* can end with the digit 0 for any natural number

*n*.

**SELF PRACTICE**: Which of the following numbers can end with the digit 0 for some natural number *n*?

i) 5^{n}

ii) 15^{n}

iii) 20^{n}

**EXERCISE**: Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

**SELF PRACTICE**: Explain why the following are composite numbers:

i) 8 × 12 × 15 + 15

ii) 11 × 12 × 13 × 14 × 15 + 13

iii) 10 × 14 × 30 + 15

iv) 21 × 22 × 23 × 24 × 25 + 12

v) 15 × 21 × 27 + 33

vi) 44 × 45 × 46 × 47 × 48 + 9

**EXERCISE**: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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