# Least Common Multiple (L.C.M) - Mathematics Form 1 Notes ## Introduction

### Definition

• LCM or LCF is the smallest multiple that two or more numbers divide into evenly i.e. without a remainder.
• A multiple of a number is the product of the original number with another number.
• Some multiples of 4 are 4, 8, 1 2, 1 6, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56 …
• Some multiples of 7 are 7, 1 4, 21 , 28, 35, 42, 49, 56 …
• A Common Multiple is a number that is divisible by two or more numbers. Some common multiples of 4 and 7 are 28, 56, 84, and 1 1 2.
• When looking for the Least Common Multiple, you are looking for the smallest multiple that they both divide into evenly. The least common multiple of 4 and 7 is 28.

Example

Find the L.C.M of 8, 12, 18 and 20. (using tables)

Solution

 2 8 12 18 20 2 4 6 9 10 2 2 3 9 5 3 1 3 9 5 3 1 1 3 5 5 1 1 1 5 1 1 1 1 1
• The L.C.M is the product of all the divisions used.
• Therefore, L.C.M. of 8, 12, 18 and 20 = 2 x 2 x 2 x 3 x 3 x5
= 23 x 32x 5
= 360

Note;

• Unlike the G.C.D tables, if the divisor /factor does not divide a number exactly, then the number is retained, e.g., 2 does not divide 9 exactly, therefore 9 is retained .
• The last row must have all values 1 . ## Past KCSE Questions on the Topic

1. Find the L.C.M of x2 + x, x2 – 1 and x2 – x .
2. Find the least number of sweets that can be packed into polythene bags which contain either 9 or 1 5 or 20 or 24 sweets with none left over.
3. A number n is such that when it is divided by 27, 30, or 45, the remainder is always 3. Find the smallest value of n.
4. A piece of land is to be divided into 20 acres or 24 acres or 28 acres for farming and Leave 7 acres for grazing. Determine the smallest size of such land.
5. When a certain number x is divided by 30, 45 or 54, there is always a remainder of 21 . Find the least value of the number x .
6. A number m is such that when it is divided by 30, 36, and 45, the remainder is always 7. Find the smallest possible value of m

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