Prime Factorisation Questions with Solutions | Testbook

Prime factorisation means breaking a number into a product of prime numbers. This page includes a collection of prime factorisation questions along with their step-by-step solutions. These examples will help students understand how to solve such problems easily. By practicing these questions, students can become more confident and improve their performance in school tests and competitive exams. Learning how to do prime factorisation is a useful math skill, and these solved questions make it simple to understand and apply in different situations.

Prime Factorisation Explained:

Prime numbers are special numbers that can be divided only by 1 and themselves. This means they have exactly two factors. Some examples of prime numbers are 2, 3, 5, 7, 11, 13, and 17.

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Factorisation means breaking a number into smaller numbers that multiply to give the original number. When we do this using only prime numbers, it is called prime factorisation.

In simple words, prime factorisation is the process of writing a number as a product of prime numbers.

For example, the number 20 can be written as 2 × 2 × 5. All three numbers (2, 2, and 5) are prime, so this is the correct prime factorisation of 20.

Note: Every number has only one unique set of prime factors. And in prime factorisation, you must use only prime numbers, no other types of numbers are allowed.

Further reading: How to find prime factors?

1. Division Method

In this method, we divide the number by the smallest possible prime number and continue dividing the quotient until we reach 1.

Steps:

1. Start dividing the number by the smallest prime (2).
2. Continue dividing until the number is no longer divisible by that prime.
3. Move to the next prime number (3, 5, 7, etc.).
4. Repeat until the final quotient is 1.
5. The prime factors are all the divisors used.

Example:
Prime factorisation of 60:

60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

So, Prime Factors = 2 × 2 × 3 × 5 = 2² × 3 × 5

 2. Factor Tree Method

In this method, we create a tree where each branch breaks the number into factors until all the end branches are prime numbers.

Steps:

1. Start with the number.
2. Break it into any two factors.
3. Keep breaking down non-prime numbers until all branches are prime.
4. The primes at the ends are the prime factors.

Example:
Prime factorisation of 60:

60

/ \

6 10

/ \ / \

2 3 2 5

So, Prime Factors = 2 × 2 × 3 × 5 = 2² × 3 × 5

Prime Factorisation Questions and Their Solutions

Now that we have a clear understanding of prime factorisation, let's put this concept into practice by solving the following problems.

Question 1: Find the prime factorisation of the following numbers:

(i) 312

(ii) 420

(iii) 7040

(iv) 6000

Solution:

(i) 312

Prime Factorisation of 312

Prime Factorisation of 312 = 2 × 2 × 2 × 3 × 13 = 2³ × 3 × 13 

(ii) 420

Prime Factorisation:420 = 2 × 2 × 3 × 5 × 7 = 2² × 3 × 5 × 7

(iii) 7040

∴ Prime factorisation of 7040 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 11.

(iv) 6000

Prime Factorisation of 6000 = 2⁴ × 3 × 5³

Question 2: Find the prime factorisation of the following numbers:

(i) 25600

(ii) 51000

(iii) 3700

(iv) 20000

Solution:

(i) 25600

Prime Factorisation of 25600

Prime Factorisation of 25600 = 2¹⁰ × 5²

(ii) 51000

Prime Factorisation of 51000

Prime Factorisation of 51000 = 2³ × 3 × 5³ × 17

(iii) 3700

Prime Factorisation of 3700

Prime Factorisation of 3700 = 2 × 2 × 5 × 5 × 37 = 2² × 5² × 37

(iv) 20000

Prime Factorisation of 20000

Final Prime Factorisation of 20000 = 2⁵ × 5⁴

Question 3: State whether the following statements are ‘true’ or ‘false’:

(i) 180 is a prime number.

(ii) 1 is a composite number.

(iii) The prime factorisation of 270 is 2 2 × 3 3 × 7.

(iv) 5 × 35 is the prime factorisation of 175.

Solution:

(i) 180 is a prime number. (False)

(ii) 1 is a composite number. (False)

(iii) The prime factorisation of 270 is 2 2 × 3 3 × 7. (False)

(iv) The prime factorisation of 175 is 5 × 5 × 7. (True)

Question 4: List the common prime factors of 288 and 168.

Solution:

Prime factorisation of 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3

Prime factorisation of 168 = 2 × 2 × 2 × 3 × 7

The common prime factors of 288 and 168 is 2 and 3.

Question 5: List the common factors of 156 and 78.

Solution:

Prime factorisation of 156 = 2 × 2 × 3 × 13

Prime factorisation of 78 = 2 × 3 × 13

The common prime factors of 156 and 78 are 2, 3, and 13.

Also Read:

• HCF and LCM
• Fractions
• Factors and Multiples
• Prime Factorisation Calculator

Question 6: Find the highest common factor of 36, 48, and 60.

Solution:

Prime factorisation of 36 = 2 × 2 × 3 × 3

Prime factorisation of 48 = 2 × 2 × 2 × 2 × 3

Prime factorisation of 60 = 2 × 2 × 3 × 5

The highest common factor of 36, 48, and 60 is 2 × 2 × 3 = 12.

Question 7: Find the lowest common multiple of 50, 80, and 130.

Solution:

Prime factorisation of 50 = 2 × 5 × 5

Prime factorisation of 80 = 2 × 2 × 2 × 2 × 5

Prime factorisation of 130 = 2 × 5 × 13

Lowest common multiple of 50, 80 and 130 = 2 × 2 × 2 × 2 × 5 × 5 × 13 = 5200.

Question 8: Fill in the blanks:

(i) 462 = 2 × ___ × ____.

(ii) 378 = 2 × __ × 3 × ___ × ____.

(iii) 1350 = 2 __ × 3 __ × 5 __ .

Solution:

(i) 462 = 2 × 3 × 7 × 11.

(ii) 378 = 2 × 3 × 3 × 3 × 7.

(iii) 1350 = 2 1 × 3 3 × 5 2 .

Question 9: Find the prime factorisation of 126000.

Solution:

The prime factorisation of 126000 is 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5 × 7.

Question 10: Verify:

Prime factorisation of 300 × Prime factorisation of 500 = Prime factorisation of (300 × 500)

Solution:

Prime factorisation of 300 = 2 × 2 × 3 × 5 × 5

Prime factorisation of 500 = 2 × 2 × 5 × 5 × 5

Prime factorisation of (300 × 500) = prime factorisation of 150000

= 2 × 2 × 2 × 2 × 3 × 5 × 5 × 5 × 5 × 5

Clearly,

Prime factorisation of 300 × Prime factorisation of 500 = Prime factorisation of (300 × 500).

Practice Questions on Prime Factorisation

1. Find the prime factorisation of the following:

(i) 2400

(ii) 500

(iii) 3700

(iv) 420

(v) 200

2. Find the common prime factors of the following:

(i) 30, 70, and 60

(ii) 144 and 360

(iii) 40, 50, and 60

3. Find the lowest common multiple of the following:

(i) 36 and 48

(ii) 50, 25, and 20

(iii) 30, 20, and 18.

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