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# What is a number divisable by?

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I've got a new idea for a continuing thread.
How can you tell what a number is divisable by. (with no remainder)

How do you tell if a number is divisable by 10 ?
ex, which of these is divisable by 10 and why?

36, 390, 487, 243000 ?
Jan 12, 2014

#1
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Trick question because any number in base ten is divisible by ten, because base ten is the basis of our numbering system. All you do is move the decimal point (called decimal point due to its ability to change a number by a factor of ten) to the left if you're dividing, and to the right if you're multiplying. If your number ends in zero, then dividing by ten will give you a remainder of zero. If your number IS a zero, then your remainder is your result and vice versa. Dividing a number by ten in base ten will always give you a remainder equal to the last digit in the number you are dividing.

To know if a number is divisible by another, first you check to see whether or not your divided number is a prime number. If it is, then you can only divide that number by itself or by 1. *Notice the reciprocity the of numbers: PrimeNumber / 1 = PrimeNumber and PrimeNumber/PrimeNumber=1

If the divided number is not a prime number, then look at the last digit of the number you are trying to divide, and note if it's an odd or even number. If its even, then you know it is absolutely divisible by 2 and will continue to be until you reach a prime number. If it's odd, then it's divisible by a prime number or by a product number of two prime numbers.

I found a remarkable sequence dividing prime numbers by 3 and got lost in wonder, and now I'm going to bed. So your best bet in knowing if a number is divisible by another is by memorizing prime numbers and their multiplication table.

Good night Everybody!
Jan 12, 2014
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WoW Zamarroncis
I actually didn't understand all of what you were trying to tell me but you sound like a budding mathematician!!

It wasn't intended to be a trick question.
I wanted to know which numbers could be divided by 10 leaving no remainder.
and, you did point out that if a number has a Zero on the end then it is exactly divisable by 10.
That's the answer I was after!
Now,
Even numbers are 0,2,4,6 and 8
you also pointed out that if a number ends in an even digit then it is divisable by 2.
great now we have 2 easy ones
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What about 5, how do we tell if a big number is exactly divisable by 5? (We don't want to do the actually division)
Jan 12, 2014
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I like three... if you add all the digits in a number and the sum is divisible by three then the number is divisible by 3
Jan 12, 2014
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Melody:

WoW Zamarroncis
I actually didn't understand all of what you were trying to tell me but you sound like a budding mathematician!!

It wasn't intended to be a trick question.
I wanted to know which numbers could be divided by 10 leaving no remainder.
and, you did point out that if a number has a Zero on the end then it is exactly divisable by 10.
That's the answer I was after!
Now,
Even numbers are 0,2,4,6 and 8
you also pointed out that if a number ends in an even digit then it is divisable by 2.
great now we have 2 easy ones
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What about 5, how do we tell if a big number is exactly divisable by 5? (We don't want to do the actually division)

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A number is divisible by 5 if it ends in 5 or 0.

*Note: 10/10=1 means dividing a number by 10 in base 10 requires the number to be multiplied by 1 before dividing by 10 and moving the last digit over one decimal place to the right of the decimal point. 10/5=2 means dividing a number by 5 in base ten requires the number to be multiplied by 2 before dividing by 10 and moving the last digit over to the right of the decimal point. 10/3=3+1/3 means dividing a number by 3 in base 10 requires the number to be multiplied by 3 then have 1/3 of the original value added to the multiplication before dividing by 10, shifting the decimal point one place to the left, which is the same as sending the last digit to the right of the decimal point. 10/4=5/2=2+1/2 means the same thing as above in base ten. This pattern is what makes all numbers divisible by ten in base ten. e.g. 10/7=1+3/7=7/7+3/7, 10/2=5, 10/6=5/3=1+2/3. web2.0calc is so handy...
Jan 12, 2014
#5
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I think I found a general complicated formula to check the divisibility of a number (x) by another (y) in base 10.

if x+(x*(10-y)/y) is divisible by 10, then x is divisible by y.
Jan 12, 2014
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Great work Doc, I like 3 too
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Now Zamarronics, are you trying to explode my brain?
Are you and our guest the same person?
I am feeling very mathed out at present but I do intend to have a proper look at what you have posted.

Melody.
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Hello again Zamarronics
I have now looked at all your post. I am going to assume that you and our guest are one and the same.
A prime no. can only be divided by 1 and itself. Definitely correct
In the paragraph below this you have not recognised the fact that 2 is a prime number.
You stated that the web2.0calc is so handy and I would like to know why, what do you mean
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Now for your formula, I looked at it fairly carefully and I think that it is completely correct. Excellent Work

Zamarronics's formula for divisibility.JPG
Jan 13, 2014
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*
Ok So what do we have so far,

1) we have Zamarronics's formula,

2) A number is exactly divisable by 10 if it has a zero at the end. (the last digit is zero)

3) A number is exactly divisable by 2 if it is even.

4) If a number is prime then it is only exactly divisable by 1 and itself

5) A number is exactly divisable by 5 if it ends in 0 or 5

6) A number is exactly divisable by 3 if the sum of its digits is exactly divisable by 3.

Doc, perhaps you could add a little to this one, it needs a couple of examples because people might not know what you mean.
Jan 13, 2014
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First of all, Zamarronics is spelled(spelt) with 'i-c-s'.

Secondly, I did post as guest, thank you for noticing.

Thirdly, I used web2.0calc.com to derive, prove, and use the formula for divisibility check.

Last, but not least, I believe an example of what 'Doc' stated goes as follows:

e.g.1 2,613 / 3 = 871 | 2+6+1+3 = 12 : 12 is divisible by 3 | Also, 12=> 1+2 = 3 since in numerology you only use numbers 0-9

e.g.2 Any number multiplied by 3 will result in a number whose digits add up to a number divisible by 3.
3 x 17 = 51 | 5+1 = 6 = 3 x 2 = 3^(2) x 2/3.
3 x 29 = 87 | 8+7 = 15 | 1+5 = 6 etc...
Jan 13, 2014
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Thank you Zamarronics, I think i have fixed all the places where I spelt your name incorrectly.
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Now for the divisability checks. What do we have so far?

1) We have Zamarronics's formula,

2) A number is exactly divisable by 10 if it has a zero at the end. (the last digit is zero)

3) A number is exactly divisable by 2 if it is even.

4) If a number is prime then it is only exactly divisable by 1 and itself

5) A number is exactly divisable by 5 if it ends in 0 or 5

6) A number is exactly divisable by 3 if the sum of its digits is exactly divisable by 3.
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There are lots more to do but lets look at these ones.
Who can give me a number that is at lease 6 digits that is
divisable by 10, 5, 3 and 2

You are not allowed to use you calculator and i don't want you to do the division.
Just with minimal work, I want you to give a number that fits the description and tell everyone why it fits the description.

There are an infinite number of answers so more than one reply would be excellent.
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I'll start. My number is 387,420
It is divisable by 10 and 5 because it ends in a 0
The last digit, 0, is even, so it is divisable by 2
3+8+7+4+2+0 = 24
and 24 is divisable by 3 (24 divided by 3=8)
Therefore 387,420 is divisable by 3
Jan 16, 2014
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How can you tell if a large numberis exactly divisable by 9
*
Please explain and give a couple of examples.
(Don't actually do the division and you don't need a calculator.)
Jan 17, 2014
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Jan 22, 2014
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Well, if I have a big number like 234, it is the same as 2x100 + 3x10 + 4 = 2x99 + 3x9 + 2 + 3 + 4.
The part 2x99 + 3x9 can obviously be divided by 9.
So if we can also divide 2 + 3 + 4 (the sum of the digits) by 9, the whole number can be divided by 9.
Jan 23, 2014
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Guest:

Well, if I have a big number like 234, it is the same as 2x100 + 3x10 + 4 = 2x99 + 3x9 + 2 + 3 + 4.
The part 2x99 + 3x9 can obviously be divided by 9.
So if we can also divide 2 + 3 + 4 (the sum of the digits) by 9, the whole number can be divided by 9.

Thankyou guest,
That is excellent.
So
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Now for the divisability checks. What do we have so far?

1) We have Zamarronics's formula,

2) A number is exactly divisable by 10 if it has a zero at the end. (the last digit is zero)

3) A number is exactly divisable by 2 if it is even.

4) If a number is prime then it is only exactly divisable by 1 and itself

5) A number is exactly divisable by 5 if it ends in 0 or 5

6) A number is exactly divisable by 3 if the sum of its digits is exactly divisable by 3.

7) A number is exactly divisable by 9 if the sum of its digits is exactly divisable by 9

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NEW QUESTION
Now, what about 4 how can we tell easily is a big number is exactly divisable by 4?
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Jan 23, 2014
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Ok so I'm not an expert with explanations but I'll try to explain myself.

Ok so if we count up 10 random multiples of 4 like 24,48,12,16,20,96,28,32,36,40. Note that every time 4, 8, or 0 is the last number, the preceding integer is even. When 2 or 6 is the last number the preceding integer is odd. So now say we have a number like 712. Because 2 is the last number, whatever comes before it has to be an odd number which in this case is 1 and it is odd, So 712 is evenly divisible by 4. Now if this number were 722, because 2 is even, it would have to be followed be either a 0, 4,or 8, to be divisible. So 728, 724, and 720 are all divisible by 4. With this in mind someone can give you any set of numbers. Such as 9848795 (random number) because this number ends with an odd number, all you would have to change the 5 to a 2 or 6 and you have a multiple of 4.

Ok so as you can see with the multiples of four, none end in 1,3,5,7, or 9 so any big number ending with one of those is not perfectly divisible by 4. Now if it ends in 0,2,4,6 or 8 then just perform the little test and you can see if its divisible by 4. Ok so, to sum it all up,

All multiples of 4 end in 0,2,4,6, or 8. If it ends in 2 or 6, check to see if the number preceding is odd or even. If it is odd then that number is divisible by 4. If the number ends in 0,4, or 8 then check to see if the preceding number is even. If it is even then the number is divisible by 4.
So with that information is 476,850,436 divisible by 4?

P.S. Thanks for the help with my atrocity of a problem Melody.
Jan 23, 2014
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Thank you Walt,

And as for your question, you are welcome. I am lucky to have good maths friends that will help me out.
Jan 23, 2014
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Ya I'm sure there is, but this is just the way that I thought about it. I tend to overcomplify things and its not always good, but hey. It works!
Jan 23, 2014
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Walt:

Ya I'm sure there is, but this is just the way that I thought about it. I tend to overcomplify things and its not always good, but hey. It works!

Yes, I'll give you that, i am sure it works. lol
Instead of looking at the last digit and the second last digit separately, why don't you try looking at the last 2 digits just as a 2 digit number. Think about that.
Jan 23, 2014
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bump
Jan 23, 2014
#19
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If the last 2 digits of a number are a multiple of 4, the whole number is divisible by 4. For example, 5784 is divisible by 4 because the last 2 digits, 84 is a multiple of 4.
Jan 23, 2014
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Walt:

If the last 2 digits of a number are a multiple of 4, the whole number is divisible by 4. For example, 5784 is divisible by 4 because the last 2 digits, 84 is a multiple of 4.

[size=150]Excellent Walt. That is what I wanted![/size]

The reason this is so is as follows.
100 is divisable by 4 therefore any multiple of 100 is divisable by 4.
For example
1537930 = 1537900 + 30
I know that 100 is divisable by 4 therefore
1537900 must be divisable by 4
Therefore the whole great long number will be divisable by 4 if and only if the last 2 digits, 30 are divisable by 4
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The summer Olympics are feld every 4 years. It is easy to know if it is an Olympic year because the last digits of the year will be divisable by 4.
For instance, 1980 80 is exactly divisable by 4 so this was a summer olympics year
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Continuing with this type of logic can you tell me how you could tell fairly easily if a big number was exactly divisable by 8

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Now for the divisability checks. What do we have so far?

1) We have Zamarronics's formula,

2) A number is exactly divisable by 10 if it has a zero at the end. (the last digit is zero)

3) A number is exactly divisable by 2 if it is even.

4) If a number is prime then it is only exactly divisable by 1 and itself

5) A number is exactly divisable by 5 if it ends in 0 or 5

6) A number is exactly divisable by 3 if the sum of its digits is exactly divisable by 3.

7) A number is exactly divisable by 9 if the sum of its digits is exactly divisable by 9

8) If the last 2 digits of a number are a multiple of 4, the whole number is exactly divisible by 4.
Jan 24, 2014
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Hmm, if I have a big number like 3456, it can be written as 3x1000 + 456.
3x1000 and numbers with more zeroes are obviously divisible by 8.
In other words, if the last 3 digits are divisible by 8, the whole number is divisible by 8!
Yay!
Jan 24, 2014
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I like Serena:

Hmm, if I have a big number like 3456, it can be written as 3x1000 + 456.
3x1000 and numbers with more zeroes are obviously divisible by 8.
In other words, if the last 3 digits are divisible by 8, the whole number is divisible by 8!
Yay!

Yes, very clever, I like Serena.
now lets see what we have.

Now for the divisability checks. What do we have so far?

1) We have Zamarronics's formula,

2) A number is exactly divisable by 10 if it has a zero at the end. (the last digit is zero)

3) A number is exactly divisable by 2 if it is even.

4) If a number is prime then it is only exactly divisable by 1 and itself

5) A number is exactly divisable by 5 if it ends in 0 or 5

6) A number is exactly divisable by 3 if the sum of its digits is exactly divisable by 3.

7) A number is exactly divisable by 9 if the sum of its digits is exactly divisable by 9

8) If the last 2 digits of a number are a multiple of 4, the whole number is exactly divisible by 4.

9) If the last 3 digits of a number are a multiple of 8, the whole number is exactly divisable by 8.
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Okay,
Here is a number 67,960,350
Without actually doing any (full) division, and without getting out a calculator, what are some of its factors.
That is, what numbers can it divided by without getting any remainders?
If you can't think of all of them maybe you can think of just a couple.
If you give reasons that would be good too.
Jan 25, 2014
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Hi Melody

I played the logic puzzle game you told us about. I haven't figured it out yet. Its a real hard one. I made it more than 1/2 way through mostly by dumb luck. It starts over when you make one mistake.

My name is Michelle and my nick name is the same as yours its Melody . My Dad gave it to me when I was 9. Its a funny story how I got it. I like the name and some of my friends back home call me Melody. I picked Puzzled as user name because I love puzzles.

I'm glad someone solve the magic number puzzle. I wounder how long it took him. One of them took me 3 days. I probably could have done it faster if I didn't have a ton of homework.

I decided to try your new puzzle what is a number divisible by.

67960350 is divisible by

10 because of the 0 at the end (rule 2)
2 because it is even (rule 3)
It's not prime because it is even.
divisible by 5 it ends in 0
divisible by 3 if the sum of its digits is exactly divisible by 3
divisible by 9 if the sum of its digits is exactly divisible by 9
not divisible by 4 because the last 2 digits are not a multiple of 4
not divisible by 8 because the last 3 digits are not a multiple of 8.

I don't know what Zamarronics's formula is. I Googled it and all that came up was preserving love energy and other weird mumbo jumbo. No math formulas though. Does it have a different name?

I read DavidQDs divisor solution and decided to practice with it, because this is coming up in the next week or so in my class. I always read ahead in the text book.

I know your rules said not to use a calculator, but after factoring out one 2, and three 3s, and two 5s, I had 50341 left over. After dividing it by the next 10 primes I figured it might be prime so I used the calculator to test it and it is.

67960350 = 2*3^3*5^2*50341

It took me a while to do it because I was doing it backward and wrong like this.

67960350 =2*3*3*3*5*5*50341 --->1,2,2,2,4,5,5,50340

Then I thought I should have added 1 to each because I am trying to find the number of divisors a number has instead of what number has this many divisors.

67960350 =2*3*3*3*5*5*50341 --->3,4,4,4,5,6,6,50342

And that is still wrong. If not for the big number at the end I might still be doing it wrong.

That's when I thought its the exponents we count not the prime numbers.

67960350 = 2*3^3*5^2*50341 ---> 2,4,3,2 This looked much better and I remembered to add 1.

Then I multiplied the numbers to gether and got 48. So there are 48 numbers that will divide into 67960350 with
a 0 remainder. I don't know how to get the divisors though.

Now that I know there are 48 divisors I used Davids example to see if there is a smaller number that has 48 divisors.

48 = 2^4*3 =2*2*2*2*3 --> 1,1,1,1,2 ---> 2^2*3*5*7*11= 4620
48 = 2^4*3 =4*2*2*3 --> 3,1,1,2 ---> 2^3*3^2*5*7 = 2520

I stooped here because making the 2 bigger with a 4 exponent is more than the 7 it would replace.

This is so cool. I wish I could give him a big hug! I read the section in my book many times and couldn't get it. Well its an E-reader not really a book and I can print the pages if I need to. But it didn't explain it good. The \$&#*#^! battery keeps going dead too.

I read his ladder answer too. I know how to do basic Trig but I don't understand this yet. I've got a loooooog way to goooooooo. Not just in math but English too. My teacher says my writing is atrocious. I laughed because my mom uses that word. It would be worse if it wasn't for spell check.
All my life everyone told me how smart I am. Now I know how dumb I can be sometimes.

When I was little my Dad told me the sun rises in the west in Australia. It was easy to get my brothers globe and picture myself standing in Australia and tell that the sun rises in the east. But he also said the water goes down the drain counter clock wise because the earths relative rotation is reversed. That was easy to see with the globe too. My Dad likes to kid and he also knows I will look things up to make sure he's rite or prove him wrong that's usually why he said these things.

But the water swirling down a drain I don't really know. I've watched a lot of drains (i like the little tornadoes) and most go clock wise but I've seen some go counter clock wise. The thing is up here the earth spins counter clock wise so I would think the water would swirl counter clock wise here and clock wise in Australia. Most people I've asked says it is opposite but weren't sure which was which. Have you ever noticed this? Or is it folklore?

Thank you
Bye for now.

Michelle a little less Puzzled, but still curious.
Jan 26, 2014
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Thankyou for this wonderful letter Michelle,

I would like to do it justice with a proper reply but I don't think that I have enough time right now.
There are a few things that I can say straight away.

I just watched the water leave two of my sinks. Both times the water spun clockwise. Yes, I too had heard it spins in a different direction in the north a and south hemisphere. This is something one of us should google properly.

Zamarronics is one of our guests. If you look back over the earlier pages of this post you will see what it is all about. The maths was quite impressive. Zamarronics came up with a
formula and it is me that called the Zamarronics formula. So I am not surprised that you couldn't find it when you googled.

I'd like to play with the factorial idea as well. DavidQD (and my other friends) did a great job on Walt's one. Maybe we could turn it into another puzzle thread. I wish I had more time but I guess that is the case with most people in life.
I was much more active with the puzzles when school was on holidays. It is difficult now as the 'proper' questions should take priority i guess. Australia hasn't gone back to school yet plus I think more people are finding the forum all the time so it might get even busier. I do want to leave time for games and number puzzles etc. This is important to me too.

Your basic level divisibility checks were all correct I believe.

I haven't looked at what you did with the higer level factorising yet, sorry.

That's it for now I think.
Melody.
Jan 27, 2014
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I am just bumping this up because i still haven't responded properly.

By the way puzzled I really didn't intend this question to be taken so far but now you have I want to totally appreciate what you have done.
When I think i have done Puzzled's entry justice I will make another post.
That's the plan anyway. I need to really study puzzled's maths before I can fully respond.

I know that 50341 is prime I checked it on the wolfram|alpha calculator. This is a really great tool that you all should make use of.
http://www.wolframalpha.com/input/?i=factors+of+50341

67960350
As Puzzled said, it is divisable by 2,3,4,8,5 and 10 and 50 since it ends in 50
= 50 * 9 * 3 * 50341
= 5*2*5*3 3*50341
= 2*5 2*3 3*50341

Now I think the number of factors altogether will be 2*3*4*2 = 48
**this is a combination question that we could put into a probability thread - comment mainly aimed at Jedithious
Okay, now I am with you. Maybe tomorrow we can try to actually discover all the divisors.
Jan 29, 2014
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bump
Jan 30, 2014
#27
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Hi again Puzzled (and all other interested parties)

Ok, we have established that there are 48 factors of 67960350
And we know the prime factors are
= 2*5 2*3 3*50341

We were looking at finding all the 48 factors.
Puzzled and I have both played with this and decided it was all too difficult to find all of them manually
but, I 'cheated'. Here they all are AND there are 48 of them

140131 factors of 67960350.JPG
Jan 31, 2014
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bump
Feb 1, 2014
#29
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I've been learning some cool stuff!

How do you find the greatest common divisor of 2 numbers if the numbers are large and you don't want to take forever to do it?

Well, you use the Euclidean Algorithm.

This is how it is done . It is really easy when you get the hang of it.

Take a look at this.