lcm (3; 224) = ? Calculate LCM, the least common multiple of numbers. Result written as an integer and prime factorized

lcm (3; 224) = ?
Approach 1. Integer numbers prime factorization. Approach 2. Euclid's algorithm.

Approach 1. Integer numbers prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


3 is a prime number, it cannot be broken down to other prime factors;


224 = 25 × 7;
224 is not a prime, is a composite number;


* Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


Calculate the least common multiple, lcm:

Multiply all the prime factors, by the largest exponents (if any).


lcm (3; 224) = 25 × 3 × 7;



lcm (3; 224) = 25 × 3 × 7 = 672
Numbers have no common prime factors: 672 = 3 × 224.


Approach 2. Euclid's algorithm:

Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

This algorithm involves the operation of dividing and calculating remainders.


'a' and 'b' are the two positive integers, 'a' >= 'b'.


Divide 'a' by 'b' and get the remainder, 'r'.


If 'r' = 0, STOP. 'b' = the GCF (HCF, GCD) of 'a' and 'b'.


Else: Replace ('a' by 'b') & ('b' by 'r'). Return to the division step above.



Step 1. Divide the larger number by the smaller one:
224 ÷ 3 = 74 + 2;
Step 2. Divide the smaller number by the above operation's remainder:
3 ÷ 2 = 1 + 1;
Step 3. Divide the remainder from the step 1 by the remainder from the step 2:
2 ÷ 1 = 2 + 0;
At this step, the remainder is zero, so we stop:
1 is the number we were looking for, the last remainder that is not zero.
This is the greatest common factor (divisor).


Calculate the least common multiple, lcm:

Least common multiple, formula:
lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b);


lcm (3; 224) =


(3 × 224) / gcf, hcf, gcd (3; 224) =


672 / 1 =


672;


Proof for the LCM formula

Least common multiple, formula:
lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b);

Let's say that the prime factorizations of 'a' and 'b' are:


a = m × n × p, where m, n, p - any prime numbers


b = m × q × t, where m, q, t - any prime numbers


=> lcm (a; b) = m × n × p × q × t;


=> gcf, hcf, gcd (a; b) = m;


Therefore:

(a × b) / gcf, hcf, gcd (a; b) =


(m × m × n × p × q × t) / m =


m × n × p × q × t =


lcm (a; b).



lcm (3; 224) = 672 = 25 × 3 × 7


Final answer:
Least common multiple
lcm (3; 224) = 672 = 25 × 3 × 7
Numbers have no common prime factors: 672 = 3 × 224.

Why do we need the least common multiple?

In order to add, subtract or compare fractions you have to first build their denominators the same. This common denominator is nothing else than the least common multiple of fractions' denominators, also called the least common denominator, lcd.


By definition, the least common multiple of two integers, LCM, is the smallest positive integer larger than 0 that is a multiple of both.



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60 is a common multiple of the numbers 6 and 15, because 60 is a multiple of 6 and is also a multiple of 15. But there is also an infinite number of common multiples of 6 and 15.

If "v" is a multiple of "a" and "b", then all the multiples of "v" are also multiples of "a" and "b".

Common multiples of 6 and 15 are: 30, 60, 90, 120... Among them, 30 is the lowest and we say that 30 is the least common multiple, or the lowest common multiple, or the smallest common multiple of 6 and 15, abbreviated as LCM.

If e = LCM (a; b), then "e" contains all the prime factors involved in the prime factorizations of both "a" and "b", by the highest powers (exponents).

Based on this rule we can calculate the least common multiple, LCM, of the three numbers in the example below:

  • 40 = 23 × 5
  • 36 = 22 × 32
  • 126 = 2 × 32 × 7
  • LCM (40; 36; 126) = 23 × 32 × 5 × 7 = 2,520

What is a prime number?

What is a composite number?

Prime numbers up to 1,000

Prime numbers up to 10,000

Sieve of Eratosthenes

Euclid's algorithm

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