Definition of LCM

LCM stands for least common multiple. In mathematics, LCM of two numbers 45 and 21, is defined as the smallest positive integer that is divisible by both. It is written as LCM(45, 21).

Properties of LCM

  • The LCM of two or more prime numbers is exactly equal to their product.
  • LCM is associative which means LCM(45, 21) = LCM(21, 45).
  • 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 given numbers is always greater than the given numbers numbers. Eg- LCM of 45 and 21 is 315, where 315 is greater than 45 and 21.

Steps to find lcm of 45 and 21 by Listing Method

Example: Find lcm of 45 and 21 by Listing Method

  • Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900, 945
  • Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945

Hence, LCM of 45 and 21 is 315.

Steps to find LCM of 45 and 21 by Common Division Method

Example: Find lcm of 45 and 21 by Common Division Method

3 45 21
3 15 7
5 5 7
7 1 7
1 1

Hence, LCM of 45 and 21 is 3 x 3 x 5 x 7 = 315.

Steps to find lcm of 45 and 21 by Formula

Example: Find lcm of 45 and 21 by Formula

  • GCF of 45 and 21 = 3
  • LCM of 45 and 21 = (45 x 21) / 3
  • => 945 / 3

Hence, LCM of 45 and 21 is 315.

Examples

Steve spends 45 dollars every day while George spends 21 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 45 and 21.
So, LCM of 45 and 21 is 315.

Boxes that are 45 inches tall are being pilled next to boxes that are 21 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 45 and 21.
So, LCM of 45 and 21 is 315.

Sammy's company prints 45 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 45 and 21.
So, LCM of 45 and 21 is 315.

Ram and Deepika are running on a circular track. They start at the same time. They take 45 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 45 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 45 and 21.
So, LCM of 45 and 21 is 315.

Find the LCM of 45 and 21 using GCF method.

Greatest common factor or gcf of 45 and 21 is GCF(45, 21) x LCM(45, 21) = (45 x 21) / GCF(45, 21) = 315.

Find the least common multiple of 45 and 21.

Least common multiple of 45 and 21 is 315.

Find the least number which is exactly divisible by 45 and 21.

Least number which is exactly divisible by 45 and 21 is 315.