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
\begin{array}{|c|c|c|c|} \hline 2 & 120 & 180 & 660 \\ \hline 2 & 60 & 90 & 330 \\ \hline 3 & 30 & 45 & 165 \\ \hline 5 & 10 & 15 & 55 \\ \hline & 2 & 3 & 11 \\ \hline \end{array} 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
\begin{array}{|c|c|c|c|}\hline2 & 72 & 18 & 22 \\\hline3 & 36 & 9 & 11\\ \hline3& 12 & 3 & 11\\ \hline & 4 & 1 & 11\\ \hline\end{array} 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
\begin{array}{|c|c|c|c|}\hline2 & 10 & 20 & 32 \\\hline & 5 & 10 & 16\\ \hline\end{array} 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
\begin{array}{|c|c|c|}\hline2 & 60 & 84 \\\hline2 &30&42 \\ \hline 3 & 15 & 21\\ \hline & 5 & 7 \\ \hline\end{array} 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
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