Definition of LCM

LCM stands for least common multiple. In mathematics, LCM of two numbers 10 and 45, is defined as the smallest positive integer that is divisible by both. It is written as LCM(10, 45).

Properties of LCM

  • LCM follows associative property, that means LCM(10, 45) = LCM(45, 10).
  • 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 10 and 45 is 90, where 10 and 45 are less than 90.
  • The LCM of two or more prime numbers is their product.

Steps to find lcm of 10 and 45 by Listing Method

Example: Find lcm of 10 and 45 by Listing Method

  • Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450
  • Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360, 405, 450

Hence, LCM of 10 and 45 is 90.

Steps to find LCM of 10 and 45 by Common Division Method

Example: Find lcm of 10 and 45 by Common Division Method

2 10 45
3 5 45
3 5 15
5 5 5
1 1

Hence, LCM of 10 and 45 is 2 x 3 x 3 x 5 = 90.

Steps to find lcm of 10 and 45 by Formula

Example: Find lcm of 10 and 45 by Formula

  • GCF of 10 and 45 = 5
  • LCM of 10 and 45 = (10 x 45) / 5
  • => 450 / 5

Hence, LCM of 10 and 45 is 90.

Examples

A shopkeeper sells candies in packet of 10 and chocolates in packet of 45. 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 10 and 45.
So, LCM of 10 and 45 is 90.

Both the cricket team and the rugby team had games, today. The cricket team plays every 10 days and the basketball team plays every 45 days. When will both teams have games on the same day again?

Given that the cricket team plays every 10 days and the basketball team plays every 45 days, so for finding the next time when both teams will play again we need to find the LCM of 10 and 45.
So, LCM of 10 and 45 is 90.

Steve spends 10 dollars every day while George spends 45 dollars every day. What is the least number of days it will take each person to spend the same amount of money?

To find the least number of days that would be taken to be able to spend the same amount of dollars we need to find the LCM of 10 and 45.
So, LCM of 10 and 45 is 90.

Sammy's company prints 10 textbooks at a time. Daniel's company prints textbooks in sets of 45 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 10 and 45.
So, LCM of 10 and 45 is 90.

Ariel exercises every 10 days and Rubel every 45 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 10 and 45 is 90.

Find the least common multiple of 10 and 45.

Least common multiple of 10 and 45 is 90.

Find the least number which is exactly divisible by 10 and 45.

Least number which is exactly divisible by 10 and 45 is 90.