Definition of LCM

LCM stands for least common multiple. In mathematics, LCM of two numbers 30 and 7, is defined as the smallest positive integer that is divisible by both. It is written as LCM(30, 7).

Properties of LCM

  • LCM follows associative property, that means LCM(30, 7) = LCM(7, 30).
  • LCM follows commutative property, which means LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c)).
  • LCM follows distributive property, which means LCM(ab, bc, ad) = d * LCM(x, y, z).
  • The LCM of two given numbers is never less than any of those numbers. Eg- LCM of 30 and 7 is 210, where 30 and 7 are less than 210.
  • The LCM of two or more prime numbers is their product.

Steps to find lcm of 30 and 7 by Listing Method

Example: Find lcm of 30 and 7 by Listing Method

  • Multiples of 30: 30, 60, 90, 120, 150, 180, 210
  • Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210

Hence, LCM of 30 and 7 is 210.

Steps to find LCM of 30 and 7 by Common Division Method

Example: Find lcm of 30 and 7 by Common Division Method

2 30 7
3 15 7
5 5 7
7 1 7
1 1

Hence, LCM of 30 and 7 is 2 x 3 x 5 x 7 = 210.

Steps to find lcm of 30 and 7 by Formula

Example: Find lcm of 30 and 7 by Formula

  • GCF of 30 and 7 = 1
  • LCM of 30 and 7 = (30 x 7) / 1
  • => 210 / 1

Hence, LCM of 30 and 7 is 210.

Examples

Franky and Joy are running on a circular track. They start at the same time. They take 30 and 7 minutes respectively to go round once. Find at what time they will run together?

Franky and Joy are running on a circular track. They take 30 and 7 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 30 and 7.
So, LCM of 30 and 7 is 210.

A shopkeeper sells candies in packet of 30 and chocolates in packet of 7. 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 30 and 7.
So, LCM of 30 and 7 is 210.

Both the cricket team and the rugby team had games, today. The cricket team plays every 30 days and the basketball team plays every 7 days. When will both teams have games on the same day again?

Given that the cricket team plays every 30 days and the basketball team plays every 7 days, so for finding the next time when both teams will play again we need to find the LCM of 30 and 7.
So, LCM of 30 and 7 is 210.

Freddy saves 30 dollars every day while Nikki saves 7 dollars every day. What is the least number of days it will take each person to save the same amount of money?

To find the least number of days that would be taken to be able to save the same amount of dollars we need to find the LCM of 30 and 7.
So, LCM of 30 and 7 is 210.

Boxes that are 30 inches tall are being pilled next to boxes that are 7 inches tall. What is the least height in feet at which the two piles will be the same height?

To find the least height in feet at which the two piles will be at same height we will find LCM of 30 and 7.
So, LCM of 30 and 7 is 210.

Sammy's company prints 30 textbooks at a time. Daniel's company prints textbooks in sets of 7 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 30 and 7.
So, LCM of 30 and 7 is 210.

Ariel exercises every 30 days and Rubel every 7 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 30 and 7 is 210.