Page 148 - Ganit Kaushal

Prime Factorisation of a Number is Unique

We see three different ways to get the prime factorisation of 48 as 48
shown in the figure. Observe that in all the three cases, we get four
times ‘2’ and one time ‘3’.
16 × 3 12 × 4 8 × 6
It shows that there is one and only one prime factorisation, except
that the prime factors may come in different orders. Since the order 2 × 2 × 2 × 2 × 3 4 × 3 × 2 × 2 2 × 2 × 2 × 3 × 2
of multiplication of numbers does not affect the result, we can
conclude that ‘The prime factorisation of a number is unique.’ 2 × 2 × 3 × 2 × 2

Remember

1. The method of expressing a given number as a product of its factors is called the factorisation of the
given number.
2. The method of expressing a given number as a product of prime factors is called prime factorisation or
complete factorisation of the given number.
3. Prime factorisation of a given number is unique except that the prime factors may come in different
orders, whereas factorisation of a number is not unique.
4. The order of writing prime factors in prime factorisation does not matter.
5. 1 and the number itself are not included in the prime factorisation of a number.

To find the prime factorisation of a number by the division method, we need to know a few divisibility rules.

5.6 RULES OF DIVISIBILITY

Divisibility by 2

A number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 as its extreme right digit (ones place). For
example: The numbers 12, 26, 44, 70, 158, 162, etc., are all divisible by 2.

Even numbers are divisible by 2. So, a number is even if it ends in 0, 2, 4, 6 or 8.

Divisibility by 4

A number is divisible by 4 if its last two digits (i.e., ones and tens) are either 0 or the number formed by them
is divisible by 4. For example: 112 is divisible by 4 because the number formed by its last two digits is 12
which is divisible by 4. Also, 296, 12300 etc. are divisible by 4.

Divisibility by 5

A number is divisible by 5 if its last digit (ones place) is either 0 or 5.

For example: 30, 45, 75, 95, 115, 120, 130, 165, 185 are divisible by 5.

Divisibility by 8

A number is divisible by 8, if its last three digits (i.e., ones, tens and hundreds) are either 0 or the number
formed by them is divisible by 8. For example: 3184 is divisible by 8, since the number formed by the last
three digits, i.e., 184 is divisible by 8. Similarly, 2640, 1496, 93000 are divisible by 8.

146 Mathematics-6