Hi everyone!
My group basically had two questions in mind:
(1) Can we divide by composite numbers when using the Ladder method?
(2) Will the order of division affect the final answer for LCM when using the Ladder method.
For efficiency sake, some of us will do a ‘short cut’ when doing the Ladder method by dividing composite number that is common to at least 2 of the numbers instead of dividing by prime numbers. This is a mentality that is common to students as well.
However, my group vaguely remembered that doing the sums this way will result in an answer bigger than the LCM possible. Thus this led us wonder if we can divide using composite numbers in Ladder method and that perhaps the order of division is important when using the Ladder method. To answer our hypothesis, we begin to dig through the different sums and here is an example we found!
Find the LCM of 60, 75 and 100.
We learnt that when finding LCM using the ladder method, we have to divide it by a factor that is common to at least 2 of the numbers.
In this question, 5 is a factor common to all the three numbers. 10 is also a factor that is common to 2 of the numbers. Both division by 5 or 10 satisfy the procedure’s requirement as stated above. Hence, we should technically get the same answer if we begin by division of 5 or 10.
Method 1: Begin by dividing by a factor that is common to only 2 numbers
10 | 60, 75, 100 |
5 | 6 , 75, 10 |
3 | 6 , 15, 2 |
2 | 2 , 5 , 2 |
1 , 5 , 1 |
Therefore, LCM = 10 x 5 x 3 x 2 x 5 = 1,500
Method 2: Begin by dividing by a factor that is common to all the factors
5 | 60, 75, 100 |
4 | 12, 15, 20 |
5 | 3 , 15, 5 |
3 | 3 , 3 , 1 |
1 , 1 , 1 |
Therefore, LCM = 5 x 4 x 5 x 3 = 300
Comparing the answer of method 1 and 2, we see that we have multiplied an additional factor of 5 in method 1, resulting in a bigger answer.
Explaining the difference
Prime factorization method:
75 = 3 x 5 x 5
100 = 2 x 2 x 5 x 5
Therefore, LCM = 2 x 2 x 3 x 5 x 5= 300
Let’s try to understand the reason for the difference in method 1 by examining the effects of our steps used on its prime factors.
If we are using method 1, we begin by division of 10.
60 = 2 x 2 x 3 x 5
75 = 3 x 5 x 5
Significance of finding
2 significances are surfaced from our above observations.
1. Order of division is important when applying ‘short cut’ of division by composite number
Suppose we are asked to find the LCM of P1, P2 P3 …Pn, division by composite number will only work if the user starts by dividing by a factor that is common to all the n numbers. Only when there are no more factors common to all the n numbers should we divide by a factor that is common to (n-1) number and so forth.
Coming back to the example of LCM of 60, 75 and 100, this would mean that we should first divide by 5 instead of 10. Cause 5 is a factor that is common to all the numbers. The numbers will then be reduced to 12, 15 and 20. We know that there are no more factors that are common to all the numbers in this set, so we should divide by the next factor that is common to 2 of the numbers. Factors common to 2 of the numbers at this stage are 2, 4 and 5. Hence, we can divide using composite number 4 because all the other factors are also common to maximum of only 2 numbers.
5 | 60, 75, 100 |
4 | 12, 15, 20 |
5 | 3 , 15, 5 |
3 | 3 , 3 , 1 |
1 , 1 , 1 |
In the end, this “short cut” requires more strategizing and may not be such an efficient method after all the planning required. This should only be recommended to students who are advance.
2. Order of division is not important when we divide using only prime numbers
Dividing by prime numbers is a sure way of getting to the right answer though sometimes it is a tedious process. Since all numbers greater than 1 are actually product of prime factors, division by prime numbers ensures that we don’t accidentally match of part of the prime factors that is also common to others (like the case that happens when we divide by composite numbers). This is the beauty of the Fundamental theorem of arithmetic.
This is a straight-forward fool proof method highly recommended to all!
After going one big round, now I finally understand why my teachers always emphasised on dividing with only prime numbers when using the Ladder method. -___-''
That is all from my group now!
Group A
P.S.: If you are interested, you can also try finding the LCM of 120, 180 and 24. If not done correctly, it will also result in an answer bigger. =)
Method 2 will certainly be a more systematic way to derive the LCM. Tedious as it might seen but students will be able to master the technique pretty soon. Depending on the perspective undertaken by students, they may derive the correct answer using different methods. That's what makes Math interesting! On the other hand, method 1 is prone to error and students need to have a strong understanding in factor to be able to employ the 'shortcut' method. However, method 1 is useful to let students have a deeper grasp of the concept and can be presented to student to stimulate deeper exploration of the topic. That's my two cents' worth :) - Wei Chuan
ReplyDeleteThank you for the insight Shi Wei! It seems that it would be better to explain HCF and LCM using Dina's Venn diagram and using prime factorisation as a proper method in obtaining them. As a beneficiary of the local education system, I've been solving for HCF and LCM using the ladder method and I remembered my teacher telling to only divide by prime numbers, hence making it foolproof. However I never understood why I'm doing the division until I realized it much later after secondary school. It came like a revelation. Back to this, I feel that ladder method can be introduced but shouldn't be placed in great emphasis as it may be prone to error like what Matthias have said.
ReplyDeleteGreat post Shi Wei! Just a few notes:
ReplyDelete1) some typo or careless mistake in your prime factorisation of 100.
2) Shi Han's point about using the diagrammatic representation can be combined with your matching off explanations for more clarity.
3) The diagrammatic representation can also be useful in constructing such questions with likely students' misconception.
Good Job!
What is the LCM ladder for 3;4;9 and 2;3;6
ReplyDeletewat is the lcm ladder for 4,12,and 16
ReplyDelete