I'm not sure about the method for finding LCM using Venn diagram. Why didn't what I did work for finding the LCM of 68 and 96?

Question

It seems that using Venn diagrams for finding the LCM of two numbers using Venn diagrams only works for some numbers, not all. Can someone please clarify? For example, for finding the LCM of 12 and 20 I use the prime factors of both, eliminating duplicates in the middle of the Venn diagram so that it's 2 x 2 x 3 x 5 = 60, which is correct. But when I did the same method for 68 and 96 it didn't work. The number obtained with this method was only in one of the times tables for those numbers.

Answer 1

You are correct that using Venn diagrams for finding the LCM of two numbers only works for some numbers and not all. Venn diagrams are a useful tool for visualizing the relationship between sets, but they have limitations when it comes to finding the LCM of numbers.

The method you used to find the LCM of 12 and 20 using prime factors and a Venn diagram is a valid method that works for most numbers. However, when it comes to finding the LCM of two larger numbers like 68 and 96, the Venn diagram method may not work as well.

One reason for this is that the Venn diagram method relies on identifying all the prime factors of the two numbers and then finding their intersection and union. For smaller numbers, this can be done easily, but for larger numbers, it can be more difficult to identify all the prime factors.

Another reason is that the Venn diagram method does not guarantee that the resulting number will be the LCM. In some cases, it may give you a multiple of the LCM, but not the actual LCM itself. This is what you experienced when you used the Venn diagram method to find the LCM of 68 and 96.

A more reliable method for finding the LCM of two numbers is to use the prime factorization method you mentioned, but instead of using a Venn diagram, you can find the LCM by multiplying the highest power of each prime factor. For example, for 68 and 96, the prime factorization is:

68 = 2^2 x 17 96 = 2^5 x 3

To find the LCM, you multiply the highest power of each prime factor:

LCM = 2^5 x 3 x 17 = 1632

This method will always give you the correct LCM, regardless of the size of the numbers you are working with.

Answer 2

I find it very weird looking for LCM using Venn diagrams, but why not?

Let's take all multiples of $68$ from $68$ itself up to $96 \cdot 68$, this is the list we get:

$V_{68} = \{68, 136, 204, 272, 340, 408, 476, 544, 612, 680, 748, 816, 884, 952, 1020, 1088, 1156, 1224, 1292, 1360, 1428, 1496, 1564, 1632, 1700, 1768, 1836, 1904, 1972, 2040, 2108, 2176, 2244, 2312, 2380, 2448, 2516, 2584, 2652, 2720, 2788, 2856, 2924, 2992, 3060, 3128, 3196, 3264, 3332, 3400, 3468, 3536, 3604, 3672, 3740, 3808, 3876, 3944, 4012, 4080, 4148, 4216, 4284, 4352, 4420, 4488, 4556, 4624, 4692, 4760, 4828, 4896, 4964, 5032, 5100, 5168, 5236, 5304, 5372, 5440, 5508, 5576, 5644, 5712, 5780, 5848, 5916, 5984, 6052, 6120, 6188, 6256, 6324, 6392, 6460, 6528\}$

Now all the multiple from $96$ from $96$ itself up to $68 \cdot 96$:

$V_{96}= \{96, 192, 288, 384, 480, 576, 672, 768, 864, 960, 1056, 1152, 1248, 1344, 1440, 1536, 1632, 1728, 1824, 1920, 2016, 2112, 2208, 2304, 2400, 2496, 2592, 2688, 2784, 2880, 2976, 3072, 3168, 3264, 3360, 3456, 3552, 3648, 3744, 3840, 3936, 4032, 4128, 4224, 4320, 4416, 4512, 4608, 4704, 4800, 4896, 4992, 5088, 5184, 5280, 5376, 5472, 5568, 5664, 5760, 5856, 5952, 6048, 6144, 6240, 6336, 6432, 6528\}$

The smallest number, being in both Venn diagrams ($\min(V_{68} \cap V_{96})$), is 1632.

Edit:

Apparently, this is not the way it should be done: the Venn diagrams should be defined, based on the prime factors, which would give following Venn diagrams:

$V_{68} = \{2, 2, 17\}$
$V_{96} = \{2, 2, 2, 2, 2, 3\}$

Hence:
$V_{68} \cap V_{96} = \{2, 2\}$, meaning that the greatest common divisor equals 4.
$V_{68} \cup V_{96} = \{2, 2, 2, 2, 2, 3, 17\}$, meaning that the lowest common multiple equals $2^5 \cdot 3 \cdot 17$, being 1632.