Cake/Ladder Method For LCM and GCD

The Cake Ladder Method, also known as the Ladder Method or the Factorization Box Method, is a technique used to find the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of two or more numbers.
The Cake/Ladder Method involves dividing the numbers with a prime number which evenly divides two or more numbers in the row.

Steps to find LCM using the Cake/Ladder method

Step 1: Write down the numbers in the row.

• Start by writing the numbers you want to find the LCM in the topmost row.

Step 2: Divide the numbers by a prime number which divides at least two or more of the numbers in the row.

• Write the prime factor to the left of the row.
• Divide each divisible number by this prime factor and write the results in the next row.
• If a number is not divisible by the chosen prime, simply bring it down to the next row unchanged.

Step 3: Continue dividing the row by prime numbers.

• Repeat the process with subsequent rows, dividing by prime factors that divide at least two numbers in the row.

Step 4: Stop when there are no common factors that can divide two or more numbers in the row.

Finding GCD using the Cake/Ladder Method

The GCD can be calculated using the Cake/Ladder Method by observing the left side of the ladder.

Step 1: Write the numbers in a row

• Place all the numbers side by side.

Step 2: Divide by common factors

• Find the smallest prime number that divides all the numbers.
• Write it on the left and divide each number.
• Repeat this step only with factors that divide all the numbers.

Step 3: Stop when no common factors remain

• Continue dividing until the numbers have no common factors other than 1.

Step 4: Multiply the left-side divisors

• The product of all the divisors used to divide all the numbers is the GCD.

Solved Examples on Finding LCM and GCD by Ladder Method

LCM by Ladder Method

Example 1: Find the LCM of 180, 120, and 660 using the Cake/Ladder method.

Solution:

LCM of 180, 120, 6600

Since, there is no other number which evenly divides two or more numbers, we stop the process.

Hence,
LCM of 180, 120, and 6600 is 2 × 2 × 3 × 5 × 3 × 2 × 110 = 19800

Example 2: Find the LCM of 72, 18, and 22 using the Cake/Ladder method.

Solution:

LCM of 72, 18, and 22

Since, there is no other number which evenly divides two or more numbers, we stop the process.

Hence,
LCM of 72, 18, and 22 is 2 × 3 × 3 × 4 × 11 = 792

GCD by Ladder Method

Example 1: Find the GCD of 10, 20, 32 using the Cake/Ladder method.

Solution:

GCD of 10, 20, 32

Since, there is no other number which evenly divides all the numbers, we stop the process.

Hence,
GCD of 10, 20, 32 = 2

Example 2: Find the GCD of 60, 84 using Ladder method.

Solution:

GCD of 60, 84

Since, there is no other number which evenly divides all the numbers, we stop the process.

Hence,
GCD of 60 and 84 = 2 × 2 × 3 = 12

Related Reads

• LCM.
• LCM Formula.
• Interesting Facts about LCM.