# Problems on LCM

## Problems on LCM Shortcut Tricks

**Common Multiple: **Defined as number which is exactly divisible by each of the numbers under consideration. **Example **60 is a common multiple of 2,3,4,5,6,10,15,20 and 30

**Lowest Common Multiple(LCM): **LCM of two or more numbers is the least or smallest number which is exactly divisible by each of them.

### Find out LCM of 12, 16

## Solution

### Methods of Finding LCM

### Method of Prime Factors

**Step-1**Resolve each of the given numbers into prime factors**Step-2**Find the product of the highest powers of all the factors that occour in the resolution of the given number. This product will be required LCM

### Find the LCM of 8, 12, 15 and 21

## Solution

8= 2*2*2= 2^3 12= 2*2*3= 2^2*3 15=3*5 21=3*7 So Product of prime numbers with highest powers (2^3)*3*5*7=840 required LCM

### Short Cut Method

Write down the given numbers in a line separating them by commas. Divide by any one of prime numbers 2, 3,5,7 which al least exactly divide at least any two of the given numbers . Repeat down the process until you get a line of prime numbers. Now the product of all divisors and the numbers in the last line will be the required LCM.

### Find LCF of 12 and 15

## Solution

The multiples of 12 are : 12, 24, 36, 48, 60, 72, 84, .... The multiples of 15 are : 15, 30, 45, 60, 75, 90, .... 60 is a common multiple (a multiple of both 12 and 15), and there are no lower common multiples. Therefore, the lowest common multiple of 12 and 15 is 60.

### Finding LCM with prime factorisations

## Solution

We want to find the LCM of the numbers 60 and 72. Start by writing each number as a product of its prime factors. 60 = 2 * 2 * 3 * 5 72 = 2 * 2 * 2 * 3 * 3

To make the next stage easier, we need to write these so that each new prime factor begins in the same place:

## Solution

60 = 2 * 2 * 3 * 5 72 = 2 * 2 * 2 * 3 * 3

All the "2"s are now above each other, as are the "3"s etc. This allows us to match up the prime factors.

The highest common factor is found by multiplying all the factors which appear in both lists:

So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.