HOW YO FIND A HCF OR A LCM VERY EASILY
Answers
LCM : The least number which is exactly divisible by each of the given numbers is called the least common multiple of those numbers. For example, consider the numbers 3, 31 and 62 (2 x 31). The LCM of these numbers would be 2 x 3 x 31 = 186.
To find the LCM of the given numbers, we express each number as a product of prime numbers. The product highest power of the prime numbers that appear in the prime factorization of any of the numbers gives us the LCM.
For example, consider the numbers 2, 3, 4 (2 x 2), 5, 6 (2 x 3). The LCM of these numbers is 2 x 2 x 3 x 5 = 60. The highest power of 2 comes from prime factorization of 4, the highest power of 3 comes from prime factorization of 3 and the prime factorization of 6 and the highest power of 5 comes from prime factorization of 5.
HCF : The largest number that divides two or more numbers is the highest common factor (HCF) for those numbers. For example, consider the numbers 30 (2 x 3 x 5), 36 (2 x 2 x 3 x 3), 42 (2 x 3 x 7), 45 (3 x 3 x 5). 3 is the largest number that divides each of these numbers, and hence, is the HCF for these numbers.
HCF is also known as the Greatest Common Divisor (GCD).
To find the HCF of two or more numbers, express each number as a product of prime numbers. The product of the least powers of common prime terms gives us the HCF. This is the method we illustrated in the above step.
Also, for finding the HCF of two numbers, we can also proceed by long division method. We divide the larger number by the smaller number (divisor). Now, we divide the divisor by the remainder obtained in the previous stage. We repeat the same procedure until we get zero as the remainder. At that stage, the last divisor would be the required HCF.
For example, we find the HCF of 30 and 42.
For two numbers ‘a’ and ‘b’, LCM x HCF = a x b
HCF of co-primes = 1
For two fractions,
HCF = HCF (Numerators) / LCM (Denominators)
LCM = LCM (Numerators) / HCF (Denominators)
A natural number , greater than 1, can always be written as sum of greatest common divisor(gcd) and lowest common multiple (lcm) of two natural numbers , i.e. ,
x=gcd(a,b)+lcm(a,b).
Let’s prove it. Let x be any natural number, greater than 1, and a and b are also two natural numbers(greater than oor equal to1).
Let’s take
a=x-1 and b=1
now we find lcm of a and b ,
lcm(a,b)=a —-1
which is simply a because lcm of any natural number with 1 is the number itself.
now we find gcd of a and b
gcd(a,b)=1 —-2
since gcd of any natural number with 1 is 1 since 1 is the highest common factor of both numbers.