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Archive for the ‘Number Theory’ Category

Secrets of the Calculator, Part 2


Welcome to the second in my series of posts about the “secrets of the calculator.” In my first post on this topic I explained how the 1/x key can make it easier to input certain expressions. If you missed that one, just click here.

Today I’d like to talk about another key that is quite useful, the “EE” key.

What in the world is the EE key, and what does it do?

At first hearing, EE sounds a bit like the call of a monkey.

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How to Use the Calculator’s 1/x Key?


In my tutoring I am continually surprised by how little most students know about their calculators.

It is true that most students know the basics:  the four operations, the exponent key, the square root key, the Pi key, and maybe some trig fundamentals, like sin, cos 60 and tan. But aside from these basic keys and keystrokes, many students have little to no idea what the other keys do.

The funny thing is that there are so many keys and keystrokes that students would just “love” if they only knew about them.

So to help students out a bit here, I’m starting an occasional series whose name is just below.

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How to See Why the Divisibility Trick for 3 Works


One of my subscribers asked why the trick for divisibility for 3 actually works. [If you missed the post on that trick, go here:]

But the gist of the trick is this:  3 divides evenly into a number if 3 divides evenly into the sum of the digits of the number.

I’ll prove this trick for a three-digit number, but you’ll see why the proof applies to numbers with as many digits as you’d like.

Let’s call our three-digit number cde, where c is the digit in the 100s place, d is the digit in the 10s place, and e is the digit in the 1s place.

We can state the value of our cde number like this:

cde =  (100 x c)  +  (10 x d)  +  e

This shows that the c-part of the number is made up of 100 groups of c. If you think about it, there’s no reason we can’t re-write the value of this digit as 99 groups of c plus 1c, or just:  (99c + c).

In the same way, the d-part of cde is made up of 10 groups of d, which we can re-write as (9d + d). And of course the e-part of the number is just e.

So now we have this:

cde =  (99c +  c)  +  (9d + d)  +  e

Using the rules of jolly old algebra, we can shuffle the terms around a bit to get this:

cde =   99c + 9d + (c + d + e)

Then, adding a set of parentheses for clarity, we get this:

cde =   (99c + 9d) + (cde)

So far, so good. But what’s the point? Well, we’re just getting to that.

Let’s think a bit more about the 99c term. Factoring out a 3, we see that 99c =  3 x 33c. Since 3 is a factor of this expression, 99c must be divisible by 3. Aha, progress, right?

Factoring the d-term, we see that 9d = 3 x 3d. Again, since 3 is a factor of this expression, 9d is  also divisible by 3.

So we now know that both 99c and 9d are divisible by 3.

We can never forget the Divisibility Principle of Sums (DPS), which says:

If a number, x, divides evenly into both a and b, then it divides evenly into their sum, (a + b).

What does that mean here? It means that since 3 divides evenly into both 99c and 9d, it must divide evenly into their sum:   (99c + 9d).

And remember that our number, cde, equals nothing more than:  (99c + 9d) + (cde)

So, we’ve just found out that 3 divides evenly into the quantity in the first parentheses:  99c + 9d

So to find out if 3 divides evenly into the whole number cde, all that’s left is to find out whether or not 3 divides into what remains, the quantity in the second parentheses:  c + d + e

But guess what? c + d + e is the sum of the digits for our number, cde. So this idea right here is the trick for divisibility for 3:  To find out if 3 divides evenly into a number, just add up the number’s digits and see if 3 goes into that sum.  If it does, then 3 does go in; if not, 3 does not go in.

So this proves the divisibility trick for a three-digit number like cde.

To see why the same trick works for numbers with four or more digits, keep in mind that the larger digits can similarly all be broken up as we broke up the digits of c and d. For example, if we have a four-digit number, bcde, then the value of the leading digit b can be viewed, first, as (1,000 x b). And then this can be split apart once more, into 999b + b. That way, this four-digit number fits into the pattern of the trick. And this same kind of split up can be done for any digit whatsoever.

So if you follow the logic of this proof, you now see that the divisibility trick rests on solid logical/mathematical ground.

How to Find out if 6 Divides in Evenly – Divisibility by 6


Award numeral 6

Image via Wikipedia

So far we’ve learned fun & easy divisibility tricks for the numbers 3 and by 4. Learning these tricks helps us reduce fractions with serious speed, and it helps us perform other math operations with a lot more ease. So let’s keep the learning
process going.

[Note:  If this is the first of these divisibility blogs that you have seen, search this blog for posts about divisibility by 3 and by 4; that way you’ll get caught up with the flow of these posts.]

The trick for 5 is incredibly simple:  5 goes into any number with a ones digit of 5 or 0. That is all you need to know. Not much else to say about 5.

And here is the trick for 6:  6 divides into any number that is divisible by BOTH 2 and 3. In other words, for the number in question, check to see if both 2 and 3 go in evenly. If they do, then 6 must also go in evenly. But if EITHER 2 or 3 does NOT go into the number, then 6 definitely will NOT go in. So you need divisibility by BOTH 2 AND 3 … in order for the trick to work.

Here’s an alternative way to say this trick, a way some kids find easier to grasp:  “6 goes into all even numbers that are divisible by 3.”

EXAMPLE 1:  74 — 2 goes in, but 3 does not, so 6 does NOT go in evenly.

EXAMPLE 2:  75 — 3 goes in, but 2 does not, so 6 does NOT go in evenly.

EXAMPLE 3:  78 — 2 and 3 BOTH go in evenly, so 6 DOES go in evenly.

Notice that since the tricks for 2 and 3 are quite simple, this trick for 6 is really quite simple too. It is NOT hard to use this trick even on numbers with a bunch of digits.

EXAMPLE 4:  783,612 — 2 goes in, and so does 3, so 6 DOES go in evenly. [checking for 3, note that you need to add only the digits 7 & 8. 7 + 8 = 15, a multiple of 3, so this large number IS divisible by 3.]

Now give this a try yourself with these numbers. For each number tell whether
or not 2, 3 and 6 will divide in evenly.

PROBLEMS:
a)  84
b)  112
c)  141
d)  266
e)  552
f)  714
g)  936
h)  994
i)  1,245
j)  54,936

ANSWERS:
a)  84:  2 yes; 3 yes; 6 yes
b)  112:  2 yes; 3 no; 6 no
c)  141:  2 no; 3 yes; 6 no
d)  266:  2 yes; 3 no; 6 no
e)  552:  2 yes; 3 yes; 6 yes
f)  714:  2 yes; 3 yes; 6 yes
g)  936: 2 yes; 3 yes; 6 yes
h)  994: 2 yes; 3 no; 6 no
i)  1,245:  2 no; 3 yes; 6 no
j)  54,936: 2 yes; 3 yes; 6 yes

 

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 

Divisibility: Find out if 3 divides evenly into an integer


Quick:  What English word has 12 letters, almost half of which are are the letter “i” — well, 5 of the 12, to be exact?

Why it’s the word “D-I-V-I-S-I-B-I-L-I-T-Y” — a great thing to understand if you’re going to spend any amount of time doing math. And guess what:  virtually ALL students do a fair amount of math, so everyone would do well to master the tricks of divisibility.

With the tricks for divisibility in your command, you will have a much easier time:

—  reducing fractions
—  multiplying fractions
—  dividing fractions
—  adding and subtracting fractions
—  finding the GCF and LCM
—  simplifying ratios
—  solving proportions
—  factoring algebraic expressions
—  factoring quadratic trinomials
—  Need I say more?

I’m sure  you get the point — divisibility tricks are handy to know.

Since the tricks of divisibility are fun and interesting, too, I’ll share as many as I can think of. If, after I’m done, you know tricks I have not mentioned, feel free to share them as comments. Or, if you know any additional tricks for the numbers I’m covering, share those! It’s always fun to learn ways to get faster at math.

Today, I’ll share the trick that tells us whether or not a number is divisible by 3. Now many of you probably know  the basic trick. But even if you do, don’t skip this blogpost. For after I show how this trick is usually presented, I’ll share a few extra tricks that most people don’t know, tricks that make the basic trick even easier to use.

Here’s how the trick is usually presented.

Take any whole number and add up its digits. If the digits add up to a multiple of 3 (3, 6, 9, 12, etc.), then 3 divides into the original number. And if the digits add up to a number that is not a multiple of 3 (5, 7, 8, 10, 11, etc.), then 3 does not divide into the original number.

Example A:  Consider 311.

Add the digits:  3 + 1 + 1  = 5

Since 5 is NOT a multiple of 3, 3 does NOT divide into 311 evenly.

Example B:  Consider 411.

Add the digits:  4 + 1 + 1  =  6

Since 6 IS a multiple of 3, 3 DOES divide into 411 evenly.

Check for yourself:

311 ÷ 3 = 103.666 … So 3 does NOT divide in evenly.

But 411 ÷ 3  =  137 exactly. So 3 DOES divide in evenly.

Isn’t it great how reliable math rules are? I mean, they ALWAYS work, if the rule is correct. In what other field do we get that level of certainty?!

Corollary #1:

Now, to make the rule work even faster, consider this trick. If the number in question has any 0s, 3s, 6s, or 9s, you can disregard those digits. For example, let’s say you need to know if 6,203 is divisible by 3. When adding up the digits, you DON’T need to add the 6, 0 or 3. All you need to do is look at the 2. Since 2 is NOT  a multiple of 3, 3 does NOT go into 6,203.

So now try this … what digits do you need to add up in the following numbers? And, based on that, is the number divisible by 3, or not?

a)  5,391
b)  16,037
c)   972,132

Answers:

a)  5,391: Consider only the 5 & the 1. DIVISIBLE by 3.
b)  16,037: Consider only the 1 & 7. NOT divisible by 3.
c)   972,132: Consider only the 7, 2, 1 & 2. DIVISIBLE by 3.

Corollary #2:

Just as you can disregard any digits that are 0, 3, 6, and 9, we can also disregard pairs of numbers that add up to a sum that’s divisible by 3. For example, if a number has a 5 and a 4, we can disregard those two digits, since they add up to 9. And if a number has an 8 and a 4, we can disregard them, since they add up to 12, a multiple of 3.

Try this. See which digits you need to consider for these numbers. Then tell whether or not the number is divisible by 3.

a)  51,954
b)  62,497
c)  102,386

Answers:

a)  51,954: Disregard 5 & 1 (since they add up to 6); disregard the 9; disregard the 5 &4 (since they add up to 9). So number is DIVISIBLE by 3. [NOTE:  If you can disregard all digits, then the number IS divisible by 3.]
b)  62,497: Disregard 6; disregard 2 & 4 (Why?); disregard 9. Consider only the 7. Number is NOT divisible by 3.
c)  102,386: Disregard 0, 3, 6. Disregard 1 & 2 (Why?). Consider only the 8. Number is NOT divisible by 3.

See how you can save time using these corollaries?

Using the trick and the corollaries, determine which numbers you need to consider, then decide whether or not 3 divides into these numbers.

a)  47
b)  915
c)  4,316
d)  84,063
e)  25,172
f)  367,492
g)  5,648
h)  12,039
i)  79
j)  617
k)  924

ANSWERS:

a)  47:  Consider the 4 and 7. Number NOT divisible by 3.
b)  915:  Consider no digits. Number IS divisible by 3.
c)  4,316:  Consider the 4, 1. Number NOT divisible by 3.
d)  84,563:  Consider only the 5. Number NOT divisible by 3.
e)  71,031:  Consider the 7, 1, 1. Number IS divisible by 3.
f)  367,492:  Consider only the 7. Number NOT divisible by 3.
g)  5,648:  Consider only the 5. Number NOT divisible by 3.
h)  12,039:  Consider no digits. Number IS divisible by 3.
i)  79:  Consider only the 7. Number NOT divisible by 3.
j)  617:  Consider the 1, 7. Number NOT divisible by 3.
k)  927:  Consider no digits. Number IS divisible by 3.

From GPGCF to GCF … in two easy steps


Once you know the GPGCF, you’re two easy steps from finding the GCF.
[If you don’t know how, see my last post:  “Recent Insight on the GCF (and GPGCF)”] That’s one of the benefits of finding the GPGCF — speed in getting the GCF! Here are the short, sweet steps:

1)  Find all factors of the GPGCF, and list them from largest to smallest.

2) Starting with the largest factor and working your way down the list, test to find the first factor that goes into both numbers. The first (largest) to do so is the GCF. You can bet on it!

Example 1 (Easy):  Find GCF for 30 and 42.

1st)  GPGCF is 12. Factors of 12, greatest to least, are 12, 6, 4, 3 and 2.

2nd)  Largest factor to go evenly into 30 and 42 is 6. So 6 is GCF.

Example 2 (Harder):  Find GCF for 72 and 120.

1st)  GPGCF is 48. Factors of 48, greatest to least, are 48, 24, 16, 12, 8, 6, 4, 3 and 2.

2nd)  Largest factor to go evenly into 72 and 120 is 24. So 24 is GCF.

NOW TRY THESE —

For each pair:

1) Find GPGCF and say if it is the difference or smaller #.
2) List factors of GPGCF, greatest to least.
3) Find GCF.

a)  8 and 12

b)  16 and 40

c)  18 and 63

d)   56 and 140

ANSWERS:

a)  8 and 12
GPGCF = 4 (difference)
Factors of 4:  4 and 2
GCF = 4

b)  16 and 40
GPGCF = 16 (smaller #)
Factors of 16:  16, 8, 4 and 2
GCF = 8

c)  18 and 66
GPGCF = 18 (smaller #)
Factors of 18:  18, 9, 6, 3 and 2
GCF = 6

d)  56 and 76
GPGCF  =  20 (difference)
Factors of 20:  20, 10, 5, 4 and 2
GCF  =  4

Recent insight on the GCF (and GPGCF)


A while back I wrote a post about the GCF, and mentioned that there’s a number  related to it — a number that I call the GPGCF. “GPGCF” stands for the “Greatest Possible Greatest Common Factor.”

In short, the GPGCF is a number that sets an upper limit for the size of the GCF. I’ve seen many students struggle when searching for the GCF, seeking hither and yon for it. I had a sense that students were checking numbers that were too large. That’s what led me to try to figure out what must the the upper limit for the GCF.

If you check out that post (10/25/10), you’ll see that, for any two numbers, I said that the difference between those numbers has to be the GPGCF.

And I was correct, to a degree.

But I recently realized that my little theory needs modifying.

For while the difference between any two numbers can be the upper limit for the GCF, that difference is not the only quantity that can set an upper limit for the GCF. There’s another quantity that plays a role.

That other quantity, I recently realized, is the size of the smaller of the two numbers.

Take the numbers 8 and 24, for example.

The difference between these two numbers is 16, so I would have said that 16 is the upper limit for the GCF. But there’s actually another quantity that limits the size of the GCF, and that quantity is 8. For since the GCF of 8 and 24 must by definition fit into both 8 and 24, it must fit into 8. And common sense tells us that there’s no number larger than 8 that can fit into 8! So the size of this number — the smaller of the two numbers — also sets an upper limit for the size of the GCF.

So my revised theory about the GPGCF is this:  when you need to find the GCF for any two numbers, look at two quantities:  1) the smaller of the two numbers, and 2) the difference between the two numbers. Both of these quantities constrains the size of the GPGCF. So therefore, whichever of these is smaller IS the GPGCF. Once you’ve found the GPGCF, that makes it easier to find the actual GCF.

I know this sounds very abstract, so let’s look at a few examples to see what I’m blabbering on about.

Example 1:  What’s the GPGCF for 6 and 16?

Smaller number is 6; difference is 10.
6 and 10 both limit the size of the GCF, but
6 is less than 10, so 6 is the GPGCF.

Example 2:  What’s the GPGCF for 8 and 12?

Smaller number is 8; difference is 4.
4 is less than 8, so 4 is the GPGCF.

Example 3:  What’s the GPGCF for 30 and 75?

Smaller number is 30; difference is 45.
30 is less than 45, so 30 is the GPGCF.

Example 4:  What’s the GPGCF for 28 and 42?

Smaller number is 28; difference is 14.
14 is less than 28, so 14 is the GPGCF.

Now, let’s go one step further. From here, how do we figure out the GCF? I’ve done a bit more thinking about this, too, and I’ll share those ideas in my next post.