1.1 Introduction
In Class 9, you learned about real numbers and irrational numbers. Now, we will study two important ideas about positive whole numbers:
- Euclid’s Division Algorithm:
When you divide a number aaa by another number bbb, you get a quotient and a remainder smaller than bbb. For example, 17 divided by 5 gives quotient 3 and remainder 2. - Fundamental Theorem of Arithmetic:
Every number can be written as a unique product of prime numbers. For example, 12 = 2 × 2 × 3.
These ideas help us find the highest common factor (HCF) and understand decimals better, like when a fraction’s decimal stops or repeats.
1.2 The Fundamental Theorem of Arithmetic (Short Explanation)
Every natural number can be broken down into prime numbers multiplied together. For example,
- 12 = 2 × 2 × 3
- 30 = 2 × 3 × 5
The Fundamental Theorem of Arithmetic says:
Every composite number has a unique prime factorization, except for the order of the factors. This means no matter how you factor a number into primes, the list of prime factors is always the same (just the order can change).
For example,
12 = 2 × 2 × 3 is the same as 3 × 2 × 2.
This theorem helps us understand numbers better and is very important in math.
Example 1:
Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.
Answer:
Example 2 :
Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Answer:
Example 3:
Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
Answer:
Example 4 :
Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
Answer:
