Number Concepts Study Guide

Learner Guide

HCF and LCM

HCF of 135 and 255 is the largest possible number that divides 135 and 255 exactly without any remainder. The factors of 135 and 255 are 1, 3, 5, 9, 15, 27, 45, 135 and 1, 3, 5, 15, 17, 51, 85, 255 respectively. There are 3 commonly used methods to find the HCF of 135 and 255 - Euclidean algorithm, prime factorization, and long division.

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Read each topic in order. Every section has a short explanation, a core rule or formula, and one worked example. The goal is to make the ideas feel simple and usable.

Big idea: Number topics become much easier when you recognise patterns, know the key definitions, and show your steps clearly.

HCF of 135 and 255

Prime factorization of 135 and 255 is given by:

\[ 135 = 3 \times 3 \times 3 \times 5 \] \[ 255 = 3 \times 5 \times 17 \]

As visible, 135 and 255 have common prime factors. Hence,

\[ \text{HCF} = 3 \times 5 = 15 \] \[ \boxed{\text{HCF of 135 and 255} = 15} \]

Given that HCF (306, 657) = 9, find LCM (306, 657)

Worked example

\[ \text{Solution:} \] \[ \text{LCM} \times \text{HCF} = \text{product of two given integers} \] \[ \text{Given: HCF (306, 657) = 9} \] \[ \text{We have to find: LCM (306, 657)} \] \[ \text{Product of the numbers} = 306 \times 657 \] \[ \text{LCM} \times 9 = 306 \times 657 \] \[ \text{LCM} = \frac{306 \times 657}{9} \] \[ \text{LCM} = 34 \times 657 \] \[ \text{LCM} = 22338 \]

Given that HCF (252, 420) = 84, find LCM (252, 420)

\[ \text{Solution:} \] \[ \text{LCM} \times \text{HCF} = \text{product of two given integers} \] \[ \text{Given: HCF (252, 420) = 84} \] \[ \text{We have to find: LCM (252, 420)} \] \[ \text{Product of the numbers} = 252 \times 420 \] \[ \text{LCM} \times 84 = 252 \times 420 \] \[ \text{LCM} = \frac{252 \times 420}{84} \] \[ \text{LCM} = 3 \times 420 \] \[ \text{LCM} = 1260 \]

Given that HCF (144, 360) = 72, find LCM (144, 360)

\[ \text{Solution:} \] \[ \text{LCM} \times \text{HCF} = \text{product of two given integers} \] \[ \text{Given: HCF (144, 360) = 72} \] \[ \text{We have to find: LCM (144, 360)} \] \[ \text{Product of the numbers} = 144 \times 360 \] \[ \text{LCM} \times 72 = 144 \times 360 \] \[ \text{LCM} = \frac{144 \times 360}{72} \] \[ \text{LCM} = 2 \times 360 \] \[ \text{LCM} = 720 \]

Find HCF (252, 420)

To find HCF (252, 420), first find their prime factorizations:

\[ 252 = 2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7 \] \[ 420 = 2 \times 2 \times 3 \times 5 \times 7 = 2^2 \times 3 \times 5 \times 7 \] \[ \text{Common prime factors: } 2^2, \; 3, \; 7 \] \[ \text{HCF} = 2^2 \times 3 \times 7 \] \[ \text{HCF} = 4 \times 3 \times 7 \] \[ \text{HCF} = 84 \]

Find HCF (306, 657)

To find HCF (306, 657), first find their prime factorizations:

\[ 306 = 2 \times 3 \times 3 \times 17 = 2 \times 3^2 \times 17 \] \[ 657 = 3 \times 3 \times 73 = 3^2 \times 73 \] \[ \text{Common prime factor: } 3^2 \] \[ \text{HCF} = 3^2 \] \[ \text{HCF} = 9 \]