Calculate the least common multiple of numbers, LCM (21; 49)

lcm (21; 49) = 147;
Numbers have common prime factors.

Approach 1. Integer numbers prime factorization. Approach 2. Euclid's algorithm. Explanations below.

Approach 1. Integer numbers prime factorization:

21 = 3 × 7;
49 = 72;

Take all the prime factors, by the largest exponents.

Least common multiple
lcm (21; 49) = 3 × 72 = 147;

Least common multiple, lcm (21; 168) = ?

Approach 2. Euclid's algorithm:

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


Step 1. Divide the larger number by the smaller one:
49 ÷ 21 = 2 + 7;
Step 2. Divide the smaller number by the above operation's remainder:
21 ÷ 7 = 3 + 0;
At this step, the remainder is zero, so we stop:
7 is the number we were looking for, the last remainder that is not zero.
This is the greatest common factor (divisor).


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

lcm (21; 49) = (21 × 49) / gcf, gcd (21; 49) = 1,029 / 7 = 147;


Least common multiple
lcm (21; 49) = 147 = 3 × 72;

Least common multiple, lcm (21; 168) = ?

Final answer:
Least common multiple
lcm (21; 49) = 147 = 3 × 72;
Numbers have common prime factors.
Least common multiple, lcm (21; 168) = ?

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Tutoring: what is it and how to calculate the least common multiple LCM of integer numbers

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