1.2 Prime Factorisation

Quick Notes & Formulae

Essential concepts for solving prime factorisation problems.

Prime Factorisation

Breaking down a number into a product of prime numbers. Use factor trees or repeated division.

Index Notation

72 = 2^{3} \times 3^{2}

. Write repeated factors as powers:

2 \times 2 \times 2 = 2^{3}

HCF (Highest Common Factor)

Using prime factors: multiply the LOWEST power of each COMMON prime factor.

LCM (Lowest Common Multiple)

Using prime factors: multiply the HIGHEST power of ALL prime factors that appear.

Square Numbers

n^{2}

where

n

is an integer. Prime factorisation has EVEN powers for all primes.

Cube Numbers

n^{3}

where

n

is an integer. Prime factorisation has powers divisible by 3.

Past Paper Style Questions

Question 1: Basic Prime Factorisation

(a) Find the prime factorisation of 72 using a factor tree. [2]

(b) Write 72 as a product of prime factors in index form. [2]

(d) Find the LCM of 72 and 90 using prime factors. [2]

(e) Explain why 72 is not a square number. [1]

Question 2: Factor Trees and Index Notation

(a) Use a factor tree to find the prime factorisation of 84. [2]

(b) Write your answer in index notation. [1]

(d) Find the LCM of 84 and 126. [2]

(e) Explain why the HCF is always a factor of both numbers. [1]

Question 3: Comparing Two Numbers

The numbers 60 and 96 are given.

(a) Find the prime factorisation of 60. [2]

(b) Find the prime factorisation of 96. [2]

(d) Find the LCM of 60 and 96. [2]

(e) Verify that

H C F \times L C M = 60 \times 96

. [2]

Question 4: Larger Numbers

(a) Write 180 as a product of its prime factors. [2]

(b) Write 180 in index form. [1]

(d) Find the LCM of 180 and 240 using prime factors. [2]

(e) Find the smallest number that must be multiplied by 180 to make it a perfect square. [2]

Question 5: Given Prime Factorizations

Two numbers have the following prime factorizations:

A = 2^{3} \times 3 \times 5, B = 2^{2} \times 3^{2} \times 7

(a) Write down the value of A. [1]

(b) Write down the value of B. [1]

(d) Find the LCM of A and B. [2]

(e) Find the number of distinct factors of A. [2]

Question 6: Square and Cube Numbers

(a) Find the prime factorisation of 150 using a factor tree. [2]

(b) Write 150 in index notation. [1]

(d) Find the LCM of 150 and 210. [2]

(e) What is the smallest number that 150 must be multiplied by to make a perfect cube? [2]

Question 7: LCM and HCF Relationships

The numbers 48 and 120 are given.

(a) Find the prime factorisation of 48. [2]

(b) Find the prime factorisation of 120. [2]

(d) Find the LCM of 48 and 120 using prime factors. [2]

(e) Explain why the LCM is always a multiple of both numbers. [2]

Question 8: Problem Solving with HCF

(a) Use a factor tree to find the prime factorisation of 90. [2]

(b) Write your answer in index form. [1]

(d) Find the LCM of 90 and 150. [2]

(e) Two ropes have lengths 90 m and 150 m. They need to be cut into pieces of equal length with no remainder. What is the longest possible length of each piece? [2]

Question 9: Working Backwards

The prime factorisation of a number is given as:

2^{4} \times 3 \times 5

(a) Write down the value of the number. [1]

(b) Write down all the factors of the number. [3]

(d) Find the LCM of this number and 120. [2]

(e) Is this number a perfect square? Give a reason. [1]

Question 10: Advanced Application

Two numbers are expressed as:

P = 2^{3} \times 3^{2}, Q = 2 \times 3 \times 5

(a) Find the value of P. [1]

(b) Find the value of Q. [1]

(d) Find the LCM of P and Q using prime factors. [2]

(e) State one common factor of P and Q other than 1. [1]

(f) Bus A departs every P minutes and Bus B departs every Q minutes. If they both depart together at 9:00 am, when will they next depart together? [2]

Question 11: Prime Factorisation of 96 and 144

(a) Use a factor tree to find the prime factorisation of 96. [2]

(b) Write 96 in index form. [1]

(d) Use prime factors to find the LCM of 96 and 144. [2]

Question 12: Prime Factorisation of 132 and 198

(a) Use a factor tree to find the prime factorisation of 132. [2]

(b) Write 132 in index form. [1]

(d) Use prime factors to find the LCM of 132 and 198. [2]

Question 13: Prime Factorisation of 105 and 140

The numbers 105 and 140 are given.

(a) Find the prime factorisation of 105 using a factor tree. [2]

(b) Find the prime factorisation of 140 using a factor tree. [2]

(d) Find the LCM of 105 and 140 using prime factors. [2]

Question 14: Prime Factors of 360 and 504

(a) Write 360 as a product of its prime factors. [2]

(b) Write 360 in index form. [1]

(d) Use prime factors to find the LCM of 360 and 504. [2]

Question 15: Working from Given Prime Factors

Two numbers are given in prime factor form:

M = 2^{4} \times 3 \times 7, N = 2^{2} \times 3^{2} \times 5

(a) Find the value of

M

. [1]

(b) Find the value of

N

. [1]

M

and

N

. [2]

(d) Find the LCM of

M

and

N

. [2]

Question 16: Prime Factorisation of 210 and 330

(a) Use a factor tree to find the prime factorisation of 210. [2]

(b) Write 210 in index form. [1]

(d) Use prime factors to find the LCM of 210 and 330. [2]

Question 17: HCF and LCM of Three Numbers

The numbers 84, 126 and 210 are given.

(a) Write each number as a product of prime factors in index form. [3]

(b) Use prime factors to find the HCF of the three numbers. [2]

(d) Explain briefly how you decided which powers to use for the LCM. [1]

Question 18: Ratio of LCM to HCF

(a) Write 72 and 180 as products of prime factors in index form. [2]

(b) Find the HCF of 72 and 180 using prime factors. [2]

(d) Hence, find

\frac{LCM}{HCF}

for these two numbers. [1]

Question 19: Parameter in Prime Factor Form

A number

K

has prime factorisation

2^{a} \times 3^{2} \times 5

, where

a

is a positive integer.

(a) Find

K

when

a = 3

. [1]

(b) Write 540 as a product of prime factors in index form. [2]

K

(when

a = 3

) and 540. [2]

(d) Find the LCM of

K

(when

a = 3

) and 540. [2]

Question 20: Prime Factor Comparison

Two numbers are expressed in prime factor form:

P = 2^{5} \times 3^{2}, Q = 2^{3} \times 3 \times 5^{2}

(a) Find the value of

P

. [1]

(b) Find the value of

Q

. [1]

P

and

Q

. [2]

(d) Use prime factors to find the LCM of

P

and

Q

. [2]

(e) Find the greatest power of 2 that is a factor of both

P

and

Q

. [1]