How do we define LCM?

LCM, abbreviation for least common multiple, is defined as the smallest number that is the product of two or more numbers 12 and 21

Properties of LCM

  • The LCM of two given numbers is never less than any of those numbers. Eg- LCM of 12 and 21 is 84, where 12 and 21 are less than 84.
  • LCM is associative which means LCM(12, 21) = LCM(21, 12).
  • LCM is commutative which means LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c)).
  • LCM is distributive, which means LCM(ab, bc, ad) = d * LCM(x, y, z).
  • The LCM of two or more prime numbers is their product.

Steps to find lcm of 12 and 21 by Listing Method

Example: Find lcm of 12 and 21 by Listing Method

  • Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252
  • Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252

Hence, LCM of 12 and 21 is 84.

Steps to find LCM of 12 and 21 by Common Division Method

Example: Find lcm of 12 and 21 by Common Division Method

2 12 21
2 6 21
3 3 21
7 1 7
1 1

Hence, LCM of 12 and 21 is 2 x 2 x 3 x 7 = 84.

Steps to find lcm of 12 and 21 by Formula

Example: Find lcm of 12 and 21 by Formula

  • GCF of 12 and 21 = 3
  • LCM of 12 and 21 = (12 x 21) / 3
  • => 252 / 3

Hence, LCM of 12 and 21 is 84.

Examples

Ram and Deepika are running on a circular track. They start at the same time. They take 12 and 21 minutes respectively to go round once. Find at what time they will run together?

Ram and Deepika are running on a circular track. They take 12 and 21 minutes respectively to go round once. We need to find out at what time (minimum) they will run together again. For this we need to find the LCM of 12 and 21.
So, LCM of 12 and 21 is 84.

A shopkeeper sells candies in packet of 12 and chocolates in packet of 21. What is the least number of candies and chocolates Annie should buy so that there will be one candy for each chocolate?

The least number of candies and chocolates Annie should buy so that there will be one candy for each chocolate we need to find the lcm of 12 and 21.
So, LCM of 12 and 21 is 84.

What is the relation between LCM and GCF( Greatest Common Factor)?

Greatest common factors or gcf of 12 and 21 is GCF(12, 21) * LCM(12, 21) = (12 x 21) / GCF(12, 21) = 84.

Sammy's company prints 12 textbooks at a time. Daniel's company prints textbooks in sets of 21 at a time. According to a survey done by a committee, both companies printed the same number of textbooks last year. Find the least number of books that each company would have printed.

To find the least number of textbooks that each company could have printed we need to find the LCM of 12 and 21.
So, LCM of 12 and 21 is 84.

Mary exercises every 12 days and Nikki every 21 days. They both excercised today. How many days will it be until they excercise together again?

The problem can be solved using LCM, because we are trying to figure out the least time until they excercise together again.
So, LCM of 12 and 21 is 84.

Find the least common multiple of 12 and 21.

Least common multiple of 12 and 21 is 84.

Find the least number which is exactly divisible by 12 and 21.

Least number which is exactly divisible by 12 and 21 is 84.