# LCM and HCF Concept

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Questions based on LCM and HCF concept are asked independently in most of the competitive exams based on a reminder on dividing….etc.
In addition to formulas learning and applying tricks helps in solving most of the LCM and HCF problems. Here in this chapter, we learn complete concepts, formulas, and problems based on different models.

### What is LCM (Least Common Multiple)?

The least number which is divisible by two or more given numbers, that least number is called LCM of the numbers.
Ex: LCM of 6 and 18 is
step 1: multiples of 6 are 6,12,18,24,30,36,42,48…….and 18 are 18,36,54,72,90…………..
step 2: The least common multiple of 6 and 18 is 18.
Therefore, the LCM of 6 and 18 is 18.

### What is HCF (Highest Common Factor)?

It is also known as the Greatest common divisor (G.C.D).
The greatest number which when perfectly divides one or more given numbers then that number is called HCF of two or more given numbers.
Ex: HCF of 5,10,20 is….
step 1: Factors of 5 are 1,5   and 10 are 1,2,5,10 and 15 are 1,3,5,15.
step 2:Common factors of 5,10,15 are 1,5.
step3: Among the common factors ,the highest common factor is 5.Therefore the HCF of 5,10,20 is 5.

### There are two methods for calculating the LCM and HCF

a) Factor Method
b) Division Method

### NOTE: For LCM And HCF

• LCM of (x,y) × HCF of (x,y)=x×y
• LCM of Fractions = LCM of the Numerators/HCF of the Denominators
• HCF of Fractions = HCF of Numerators/LCM of the Denominators

### IF HCF OF a AND b IS c THEN, THE HCF OF

a) a,a+b is also c
b)a,a-b is also c
c)a+b,a-b is  also c

## Different Models in LCM and HCF Concept

### LCM MODELS:

There are three models in this LCM concept which are frequently asked in competitive exams.

#### MODEL 1:

Any number (N) which when divided by p,q or r leaving the same reminder s in each case then that number must be in the form of N=K(LCM of p,q, and r ) + s.here k=0,1,2,3…….

#### MODEL 2:

Any number (N) which when divided by p,q or r leaves respective remainders of s,t and u, where (p- s)=(q-t)=(r-u) =v, then that number must be in the form of
N=K(LCM of p,q and r)-v, here k=1,2,3,4……..

#### MODEL 3:

Any number which when divided by p and q leaves respective remainders of r and s then that number must be in the form of N=K(LCM of p and q )+ n,
where n is the smallest integer solution for the equations n=pm1 + r and n=qm2 + s here m1 and m2 are natural numbers.

### HCF MODELS:

There are three models in this HCF concept which are frequently asked in competitive exams.

#### MODEL 1:

The largest number which divides the numbers p, q and r the remainders are the same in each case then that largest number is given by the HCF of any two or three numbers (p-q), (q-r) and (p-r).

#### MODEL 2:

The largest number which divides the numbers p,q or r to give remainders of s,t and u respectively is given by HCF of any two of three numbers
(p-s), (q-t) and (r-u).