Why Prime Factorisation Always Works the Same Way

To break a number into its prime factors, keep dividing it by the smallest prime numbers until only primes are left; this is called prime factorisation, and every number has a unique prime factorisation (apart from the order of the factors). In CBSE Class 6 Mathematics, you use either a factor tree or repeated division to do this. A factor tree splits a number into two factors, then splits each factor until all branches end in prime numbers. Repeated division divides the number by the smallest possible prime repeatedly. For example, to factorise 36:
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
So, 36 = 2 × 2 × 3 × 3. No matter which method or order you use, you always get the same set of prime factors (here, two 2s and two 3s). This is called the uniqueness of prime factorisation. A common mistake is to stop at a non-prime factor or to think the order of factors changes the factorisation, but only the set of primes matters.

Prime factorisation helps you find common factors and multiples of two or more numbers, and check if numbers are co-prime. To find common factors, write the prime factors of each number and look for primes they share. For common multiples, multiply all the different primes (using the highest power from each number). Two numbers are co-prime if they have no common prime factors except 1. For example, consider 18 and 24:
- 18 = 2 × 3 × 3
- 24 = 2 × 2 × 2 × 3
Common factors: 2 and 3 (since both have these primes). The highest common factor (HCF) is 2 × 3 = 6. The least common multiple (LCM) is 2 × 2 × 2 × 3 × 3 = 72 (taking the highest powers). Now, 8 = 2 × 2 × 2 and 15 = 3 × 5 have no common prime factors, so they are co-prime. Many students think co-prime means both numbers must be prime, but any two numbers with no common prime factors are co-prime, even if neither is prime.

You can use divisibility tests to check if a number is divisible by 2, 4, 5, 8, or 10 just by looking at its digits. For 2, if the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2. For 5, if the last digit is 0 or 5, it is divisible by 5. For 10, the last digit must be 0. For 4, if the last two digits form a number divisible by 4, the whole number is divisible by 4. For 8, check the last three digits—if they form a number divisible by 8, so is the whole number. For example, 3,216: last digit is 6 (even), so divisible by 2; last two digits 16 (divisible by 4), so divisible by 4; last three digits 216 (divisible by 8), so divisible by 8. Many students mistakenly check all digits for 4 or 8, but only the last two or three digits matter.

Divisibility tests work because of place value and how numbers are built from powers of 10. For 2, 5, and 10, the last digit decides divisibility because 10 is a multiple of these numbers. For 4, 100 is divisible by 4, so only the last two digits affect divisibility by 4. For 8, 1,000 is divisible by 8, so only the last three digits matter. For example, in 2,348, the last two digits are 48 (divisible by 4), so 2,348 is divisible by 4. This is because any number can be written as (some multiple of 100) plus the last two digits, and the multiple of 100 is always divisible by 4. The same logic applies for 8 with the last three digits. Some students think you must check the whole number, but understanding place value makes these tests quick and reliable.