Fundamentals of Arithmetic: Prime Factorization

Questions and Answers

2 is a perfect square

2 is a prime number

2 is rational (correct)

2 is an integer

b2 is divisible by 2 (correct)

b is an integer

b2 is divisible by 4

c is an integer

Because a and b have a common factor other than 1

Because a and b have at least 3 as a common factor (correct)

Because a and b are integers

Because a and b are coprime

3b2 = a2 Signup and view all the answers

It is always irrational Signup and view all the answers

To show that if p divides a2, then p divides a Signup and view all the answers

5 – 3 is rational Signup and view all the answers

It is always irrational Signup and view all the answers

Because both proofs use a similar method of assuming rationality and arriving at a contradiction Signup and view all the answers

It is used to show that if p divides a2, then p divides a Signup and view all the answers

Study Notes

Any natural number can be written as a product of its prime factors.
Any natural number can be obtained by multiplying prime numbers, allowing them to repeat as many times as we wish.
There are infinitely many primes, and if we combine all these primes in all possible ways, we will get an infinite collection of numbers, which include all the primes and all possible products of primes.

HCF (a, b) × LCM (a, b) = a × b for any two positive integers a and b.
HCF (a, b) = Product of the smallest power of each common prime factor in the numbers.
LCM (a, b) = Product of the greatest power of each prime factor involved in the numbers.

Example 2: Find the LCM and HCF of 6 and 20 by the prime factorisation method.
- HCF (6, 20) = 2 and LCM (6, 20) = 2 × 2 × 3 × 5 = 60.
Example 3: Find the HCF of 96 and 404 by the prime factorisation method.
- HCF of these two integers is 22 = 4, and LCM (96, 404) = 9696.
Example 4: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
- HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6, and LCM (6, 72, 120) = 23 × 32 × 51 = 360.

A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
Examples of irrational numbers include 2, 3, π, - 0.10110111011110..., etc.
Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
Theorem 1.3: 2 is irrational.

The proof of irrationality is based on the Fundamental Theorem of Arithmetic and the technique of 'proof by contradiction'.
The proof involves assuming that the number is rational and then showing that it leads to a contradiction.

Example 5: Prove that 3 is irrational.
- Similar proof to that of 2, using Theorem 1.3 with p = 3.
Example 6: Show that 5 – 3 is irrational.
- Assume, to the contrary, that 5 – 3 is rational, and then show that it leads to a contradiction.