When a number is expressed as a product of its factors we say that the number has been factorised. Thus, when we write 24 = 3×8, we say that 24 has been factorised. This is one of the factorisations of 24. The others are :

In all the above factorisations of 24, we ultimately arrive at only one factorisation 2 × 2 × 2 × 3. In this factorisation the only factors 2 and 3 are prime numbers. Such a factorisation of a number is called a prime factorisation. Let us check this for the number 36.

The prime factorisation of 36 is 2 × 2 × 3 × 3. i.e. the only prime factorisation of 36.

Help yourself:
Write the prime factorisations of 16, 28, 38.

Activity:

Factor tree
Choose a number and write it

Think of a factor pair, say, 90=10 × 9 = 90

Now think of a factor pair of 10
10 = 2 × 5

Write factor pair of 9
9 = 3 × 3

Try this for the numbers
(a) 8
(b) 12

Example 7

Find the prime factorisation of 980.

Solution :
We proceed as follows:

We divide the number 980 by 2, 3, 5, 7 etc. in this order repeatedly so long as the quotient is divisible by that number. Thus, the prime factorisation of 980 is 2 × 2 × 5 × 7 × 7.

EXERCISE 3.5

Q1. Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

A 1.
(a) False
(b) True
(c) False
(d) True
(e) False
(f) False
(g) True
(h) True
(i) False

Q2. Here are two different factor trees for 60. Write the missing numbers.

(a)

(b)

A2.(a)

(b)

Q3. Which factors are not included in the prime factorisation of a composite number?

A3. 1 and the number itself

Q4. Write the greatest 4-digit number and express it in terms of its prime factors.

A4. 9999, 9999 = 3 × 3 × 11 × 101

Q5. Write the smallest 5-digit number and express it in the form of its prime factors.

A5. 10000, 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5

Q6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

A6. 1729 = 7 × 13 × 19
The difference of two consecutive prime factors is 6

Q7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

A7. (i) 2 × 3 × 4 = 24 is divisible by 6.
(ii) 5 × 6 × 7 = 210 is divisible by 6.

Q8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

Q9. In which of the following expressions, prime factorisation has been done?
(a) 24 = 2 × 3 × 4
(b) 56 = 7 × 2 × 2 × 2
(c) 70 = 2 × 5 × 7
(d) 54 = 2 × 3 × 9

A9. (b), (c)

Q10. Determine if 25110 is divisible by 45.
[Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].

A10. Yes

Q11. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.

A11. No. Number 12 is divisible by both 4 and 6; but 12 is not divisible by 24.

Q12. I am the smallest number, having four different prime factors. Can you find me?

A12. 2 × 3 × 5 × 7 = 210