1.4. Summary

1. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic states that any composite number can be broken down into its prime factors uniquely, meaning there is only one way to express a composite number as a product of prime numbers. The order of the factors does not matter; for example, the number 12 can be factored as 2 × 2 × 3, or 2 × 3 × 2, or 3 × 2 × 2. All of these represent the same prime factorization.

Think of a recipe for a cake. You can make a cake with certain ingredients in different sequences, but the ingredients themselves define what type of cake it is. Similarly, the prime factors define a composite number, and their unique combination (regardless of order) defines that number.

2. If p is a prime and p divides a2, then p divides a, where a is a positive integer.

This statement tells us that if a prime number p divides the square of a positive integer a (represented as a²), then it must also divide the integer a itself. For example, if we take p = 2 and a = 4, we see that 2 divides 4 (because 4 = 2 × 2), and it also divides 4's square, which is 16 (because 16 = 2 × 8).

Imagine a group of students that can each hold a certain number of books. If there are 16 books and the number of books each student can hold is 4 (a group size), then we can say that this group configuration allows for evenly distributing books. If there's a larger number of books (such as 16 books), it just means that each student is still capable of holding their allotted amount.

3. To prove that 2, 3 are irrationals.

This point indicates that the numbers 2 and 3 cannot be expressed as fractions of two integers (in simplest form). This was shown through proofs that involved assuming they could be expressed as rational numbers and arriving at contradictions. For instance, assuming that √2 is rational leads to the conclusion that both integers in the fraction must have 2 as a common factor, which contradicts our assumption of them being coprime.

Consider trying to measure the diagonal of a square with a side length of 1. If you attempt to express this length in terms of fractions (like 1/2, 1/3, etc.), you'll find that you cannot find exact fractions that equal the diagonal length. This is much like trying to find a fold in an origami that doesn’t exist, illustrating the concept of irrational numbers.

A N R OTETOTHE EADER You have seen that : HCF (p, q, r) × LCM (p, q, r)  p × q × r, where p, q, r are positive integers.

This note highlights a common misconception regarding the product of the highest common factor (HCF) and lowest common multiple (LCM) of numbers p, q, and r. While it is true that the product of two numbers equals the product of their HCF and LCM, it does not hold when it involves three or more numbers. This distinction is crucial for understanding the relationships and calculations involving multiple integers.

Think about three different pizza sizes. If you know how much topping to share among three sizes (small, medium, and large), you can calculate how much to put on each, but adding their individual toppings together won’t necessarily equal what you'd expect for a combined total unless you re-calculate for the shared portions. Thus, just multiplying the individual totals doesn’t work when combined rationally.

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