Algebra

The Least Common Multiple Function

Last time, we discussed the Greatest Common Denominator function. Now, I turn to its counterpart, the LCM. What does the LCM function mean, and how can we calculate it?

LCM: A simple explanation

The LCM function, written as lcm(a, b), finds the lowest possible number into which both a and b will divide evenly. For example, the LCM of 6 and 8 is 24 because 24 is the smallest number that is divisible by 6 and 8.

Find the least common multiple of the following pairs of numbers: 3 and 6, 9 and 12, 4 and 5.

Notice that in the last example above, we arrive at a special case when a and b are relatively prime; that is, there is no number that divides evenly into both a and b except 1. In this case, the LCM turns out to be the product of the two numbers.

LCM: A concrete example

In the last post, I gave the example of tiling a rectangular floor with square tiles to illustrate the Greatest Common Denominator. In accordance with the reciprocity of these two functions, I will now present a similar yet opposite scenario.

Consider a large, elaborately patterned rectangular tile—say its dimensions are 4 feet by 6 feet—squatting unceremoniously on an unfinished concrete floor. Mr. Baxter the contractor examines this tile along with his new clients, Mr. and Mrs. Calliope. “So you see, Mr. Baxter,” young Mr. Calliope is saying in a slightly stilted dialect of English, “we must use this exact type of tile. No other tile will do, you see. But also, you see, this room must also be square, yes? Now you see the challenge.”

Mr. Baxter remains silent. Then, like a long-forgotten arcade machine, he forlornly spits out a terse reply. “Twelve feet.” Mr. and Mrs. Calliope, who have evidently forgotten their elementary arithmetic, cast him a brief look of distaste and turn back to the masterpiece of a floor tile. “Twelve feet, I reckon,” continues Mr. Baxter, straightening his hat as though it gave him courage to keep talking. “The room will have to be twelve feet square.”

At this, Mrs. Calliope finally registers what Mr. Baxter is saying. “Ah, you see!” she exclaims. “But how, yes, how can you be so sure, Signor?”

As you may have guessed, our contrary proposition for the lowest common multiple function, in contrast to tiling a rectangular floor with square tiles, is tiling a square floor with rectangular tiles. The task is to find the smallest possible square room into which our ornate rectangular tile will fit EXACTLY.

Find the smallest room that can be tiled evenly with the following sizes of tiles: 6 by 9, 2 by 4, 8 by 10.

LCM: Two methods for finding it

In the Greatest Common Denominator post, we discussed using prime factorization to find the GCF of two numbers. Continuing our contrast, we can use a similar yet opposite method to find the LCM of two numbers.

Consider the example lcm(20, 22). The prime factorization of 20 (which you can find using a factor tree) is 5\times2^2. The prime factorization of 22 is 11\times2.

In the Greatest Common Denominator post, we took each factor with the lowest power, and omitted it if it wasn’t in both a and b. For the LCM, we’ll take each factor with the highest power, and we’ll keep it even if it isn’t in both numbers. For instance, we’ll take 5, 2 squared because its power is higher than just 2, and 11. Multiplying our new list of factors, we get an LCM of 220.

For the second method, let’s look at a close opposite to the floor tiling method we examined in the GCF post. Then, we subtracted square regions until all that was left was a square region. Now, we add rectangular regions until we have a square region. For instance, given lcm(16, 20), 16 is less so we add 16 to itself. Then between 32 and 20, we add 20 to itself because 20 is less than 32. We then have 32 and 40, 48 and 40, 48 and 60, and 60 and 60. The numbers are now equal, so 60 is the LCM.

LCM: Applications

The LCM is used in situations where, like Mr. Baxter’s predicament, we need to find where two dimensions become equal. For instance, one would use lcm(15, 20) to find out the time between two trains meeting in a station. In mathematics, the LCM is crucial to adding and subtracting fractions. For instance, the denominator of 1/6 + 1/8 is lcm(6, 8).

Try these problems using LCM. Remember that the LCM of three numbers a, b, and c is simply lcm(lcm(a, b), c).

1. Two trains leave a station at 12:00 noon. The A train visits the station every 25 minutes, and the B train visits the station every 15 minutes. At what time will the two trains meet in the station again for the first time?

2. What is the smallest integer that is a multiple of 14, 9, and 21?

I hope this article helped you understand the lowest common multiple function. Please leave a comment if you have any questions!

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Apps, Isosceles

Isosceles Featured for Middle School Math

Isosceles free has been featured for middle school math apps on the App Store! This is an awesome place for teachers to discover Isosceles as a new tool in the classroom. But unfortunately, Isosceles’s rating is sadly low…

If you use Isosceles free and it’s useful for you, please support Base 12 Innovations by leaving a good rating on the App Store page! Every rating helps convince teachers that Isosceles is the best geometry app for their students, and we need all the ratings and reviews we can get.

If you’re having problems using Isosceles, by all means contact me in the app’s help menu! I’ve received lots of feedback from users so far, and their support has directly helped me put together the past few updates.

You can rate Isosceles free here on the App Store.

Thank you so much for using Isosceles, and I hope you’ll leave a positive rating on the app page to support me and my apps!

Apps, Geometry, Isosceles

Isosceles Tip: Isometry

One little-known feature of Isosceles+ is its ability to create 3-dimensional sketches with the Isometric geometry system. If you have paid for the premium features pack, you can use this system by tapping the gear icon, then checking Isometric. You can use this feature to make 3D graphs, like this:

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You can also make cool optical illusions and designs, like this one:

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Some of the other features in Isosceles that facilitate these drawings:

  • To shade the graph, I used the Link tool, then added a fill to the resulting polygon. To link lines together and create a polygon, select the individual lines by tapping and holding on each one separately. Then, if a menu isn’t already showing, tap one of the blue lines to show the menu. If the lines form a closed polygon, the option will be available to link them together.
  • For the 3D cross, I had to use the Polygon tool to draw two 12-gons, which represent the two shaded crosses. Then I used the Points tool to move each vertex to its proper location.
  • To fill the polygons, select all the polygons which should have the same fill color by tapping and holding on their sides. In the menu that appears, tap Info. Choose Fill Color to change the color.
  • For this kind of drawing, it’s important to turn Snaps to Grid, Snaps to Points, etc. This ensures that your sketch will look precise, which is key for a drawing of this type.

As you can guess, an isometric grid feature has applications in engineering, construction, and other practical fields. Let me know how you use the Isometric geometry system in the comments!

Apps, Geometry, Isosceles

Isosceles Tip for v1.3/1.4: Architecture

I’ve just released a new version of Isosceles, version 1.3 for isosceles and 1.4 for isosceles+. The new version adds some more cool stuff:

  • You can now make text annotations bold, italic, underline, and strikethrough (device permitting).
  • Add a name to a line.
  • You can easily move a point to the nearest integer coordinates or key in a custom location in the point’s Info menu.
  • Isosceles premium users can now edit background colors, grid colors, and default pencil colors for individual sketches.
  • Many bug fixes which allow Isosceles users to create and maintain more complex designs than before (see below).

To celebrate the new addition, this tip will illustrate some of the amazing things Isosceles 1.4 can do. Take a look at this floor plan made with Isosceles:

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Some of the features I used to create this drawing:

  • To create the background color, go to Settings (the gear icon), tap Canvas, then choose Background Color and customize the color as desired.
  • To set the default pencil color to white, follow the above instructions but choose Heavy Pencil Color and/or Light Pencil Color. (You can toggle between these in your sketch by tapping the green pencil button.)
  • To create the doors, draw a line to represent the open door. Then choose the compass tool, move the center handle to the place where the hinges would go and drag the outer handle along the blue line to show the door’s path.
  • To draw the piano, simply follow the steps above to draw several consecutive arcs. The largest arc would have a center inside the piano, the next one outside, and the last one on the edge of the piano.

What makes this so easy in Isosceles is that you can zoom in as far as you need to without losing accuracy and snap points easily to gridlines. Version 1.4 allows you to style the sketch however you want very quickly. The compass tool also opens up new and innovative designs that simply can’t be done in most analytic geometry programs. And to top it all off, you can export any sketch into a crisp PDF document for print or web.

Please let me know in the comments how Isosceles 1.4 is working for you!

Algebra

The Greatest Common Denominator Function

Two of the most basic functions in number theory are the LCM (least common multiple) and GCD/GCF (greatest common denominator/factor) functions. They are opposites of each other, but not in the same way as addition and subtraction, multiplication and division, etc. What do these functions actually mean, and how can we use repetitive methods to evaluate expressions using LCM and GCD?

Greatest Common Denominator (GCD)

First let’s look at the Greatest Common Denominator function. Written as gcd(a, b), the GCD function determines the largest number by which a and b are both divisible. For instance, the GCD of 6 and 8 is 2 because 3, 4, 5, and 6 cannot be divided evenly into both 6 and 8.

Evaluate the GCD of the following pairs of numbers: 3 and 6, 15 and 20, 8 and 20, 3 and 5.

As evidenced by the last problem above, a special case occurs when the two numbers are relatively prime. In familiar terms, this means there are no numbers that can be divided into both a and b except 1. For instance, 7 and 12 are relatively prime and so their GCD is 1.

GCD: A concrete example

Mr. and Mrs. Allman, proud new homeowners, are in the rare position to choose the decor on the house they will inhabit for years to come. They roam the barren rooms with the stolid contractor Mr. Baxter, animatedly conjecturing about the new Jacuzzi, the new modern skylights, etc. But for now, Mr. Baxter brings their attention to a more mundane issue: the kitchen flooring.

“This here kitchen is 24 by 30 feet,” says Mr. Baxter in his stolidly guttural voice. “I need to know the biggest size of tiles I could cover it with.”

Mr. Allman, who happens to teach mathematics at a nearby high school, speaks up. “We could use the greatest common factor function—” Mrs. Allman gently cuts him off. Mathematics has no place in the kitchen.

No, no, let Mr. Allman speak. The GCD of 24 feet by 30 feet would in fact give Mr. Baxter the size of the largest square tile that fits (or divides) evenly into the room dimensions. A little computation tells us that the GCD returns a value of 6 feet.

Mr. Baxter grunts. “Okay, I’ll look into it,” he says grudgingly. “Me, I’ve never seen a 6-foot kitchen tile, but I’ll look into it.”

Find the largest square tile that will fit into a room with these dimensions: 9′ x 12′, 25′ x 24′, 18′ x 30′.

GCD: Two methods for finding it

So far, the only way we’ve considered is by guessing and checking—I think 2 is the GCD; no, wait, 4 is also a factor, so is 6. But what about when we hit numbers with many factors, such as gcd(420, 440)?

Note: Many will find it terrifying that I use the word “algorithm” in the following paragraphs. Please do not fear! An algorithm is simply a series of steps that can be repeated until you get the answer. The below methods are really not that complicated, and I’ve tried to explain them as best I can using that limited medium we call English.

1. The secret lies in prime factorization. To find the prime factors of a number is to find its base code, the series of ingredients from which a number is made. In fact, every integer can be written as the product of prime factors. If we find the prime factorization of 420, we find that

420=2^2\times3\times5\times7

When we prime factorize 440, we find that

440=2^3\times5\times11

The trick to finding the GCD is for each prime number we see, putting the term with the smaller power into the answer. If a prime factor isn’t in both numbers, we can’t accept it because there’s no way it will divide evenly into both numbers. For instance, we’ll take 2 squared and reject 2 cubed. We won’t take 3 because it’s not in 440’s factorization. We’ll also take 5, but not 7 or 11. Multiplying all these factors together, we get a final GCD of 20.

2. The second algorithm is a classic one used frequently by computers, but it can help you, too. Given gcd(240, 440), first determine the larger number—obviously 440. Now subtract the smaller number, 240, from this: 200. Then repeat given the difference and the smaller number: gcd(200, 240). We then arrive at gcd(40, 200), then gcd(160, 40), then gcd(120, 40), gcd(80, 40), gcd(40,40). Stop! We can’t go any further. The GCD of 240 and 440 is irrevocably 40.

In symbols, this method can be expressed much more simply. Given that a is a smaller number than b, we can declare that gcd(a, b) = gcd(ba, a). When a = b, we have our GCD.

How can we explain this method’s captivating simplicity and consistency? To do this we need to return to Mr. and Mrs. Allman, standing in the floorless kitchen, the sound of Mr. Baxter’s boots echoing on the concrete as he searches for 6-foot tiles.

“That was a fascinating application of math, dear,” says Mrs. Allman. “How on earth could you know that 6 feet is the right number?”

“Well, honey,” says Mr. Allman, falling into the lingo of his algebra class, “look at the shape of this floor: 24 feet by 30 feet. A square tile will of course fit into a square space, right?”

“Whatever you say, sweetie,” says Mrs. Allman dreamily.

“Yes, well.” Mr. Allman clears his throat and straightens his glasses. “If a square tile can fit into the a square space, we can remove a square space from the room and focus our attention on the remaining part: only 24 feet by 6 feet. Then we can remove another square space, and another, and another, and all we have left is a 6 by 6 square. And there’s the answer!” Mr. Allman seems to stand up straighter at that moment, as though the answer itself has reinforced his posture.

“That’s simply wonderful, dear,” said Mrs. Allman. “How much did you say the Jacuzzi would cost?”

Returning to the pristine world of mathematics, Mr. Allman has showed that the GCD can be found by removing square sections of a rectangle. The final square section remaining is the long-awaited answer.

Practice using both methods described above by calculating gcd(261, 378) and gcd(144, 162).

In the next post, we’ll discuss the lowest common multiple and its similarity to the GCD. If you have any questions, please leave a comment below.

Apps, Geometry, Isosceles

Isosceles Tip: Optics

In my Physical Science class, we are currently learning about mirrors, virtual and real images, etc. I realized that this is actually a great application for Isosceles, my geometry application. Here’s a diagram I drew to represent the reflection of an arrow in a concave mirror:

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The colored lines represent individual light rays from the object, the arrow. Each of the rays intersects at a point under the axis. Some of the techniques I used to create the diagram follow:

  • To create the mirror, use the compass tool and position the center handle on the desired center point. Drag the outer handle around the blue circle to draw the mirror.
  • To create the axis, open the mirror’s Info menu by tapping and holding on it, then pressing Info. Choose Construct Midpoint and Construct Center Point, then tap the Line tool and connect these two points. To extend the line, open its Info menu, then change the Extends switch to On.
  • To construct a parallel ray, use the method described here.
  • To add an arrow to a line, open its Info menu by tapping and holding on it, then pressing Info. Change the Start Arrow or End Arrow switch to On.

This is just one of the many ways you can use Isosceles’s versatile geometry engine. Leave a comment and tell us how you use Isosceles!