Relationship of 777 and 13?

[joeknockz]

Hi all, my name's Joe. I'm not educated in calculus or any college math courses. But I do love math. I play with numbers all day. Some people doodle while I just write numbers out and times them when bored instead.

Anyway, now that you know why my question might seem silly to you, here it is:

Is there a special relationship between 777 and 13?

I noticed that when I times 777 by 13 I get 10101. It was an interesting discovery to me when finding the repetitive nature of 1 and 0. And of course 777 x 13 x 2 would equal 20202. I noticed that whatever the multiplier after 777 x 13 is (between 1 and 99) you'll see that number written out 3 times. for example 777 x 13 x 8 = 80808. And 777 x 13 x 65 = 656565. And 777 x 13 x 4.7 = 47474.7

Is there a known relationship of the numbers 777 and 13? Are they used in any formulas for anything or is it just one of those interesting coincidences? Again, I'm not schooled in math, just what I've taught myself and after looking for a place my question might be welcomed i found you guys...

Joe

 

[disregardthat]

Notice that $$777=7*111$$, so $$13*777=7*13*111=91*111=(91)(100+10+1)=9100+910+91$$ which you will se have a cancelling and shifting effect, as 9+1=10.

The fact that $$10101*a = a0a0a$$ if $$0<a<10$$, or $$10101*a=aaa$$ if $$9<a<100$$, is because $$10101=10000+100+1=10^4+10^2+1.$$
You know every number can be written as it's digits muliplied with 10 to the power of the spot it has as a digit, (first spot --> 0, second spot --> 1) i.e $$563=5*10^2+6*10^1+3*10^0$$

Now if a is between 0 and 10, $$10101*a=a*10^4+0*10^3+a*10^2+0*10+1$$, which is naturally is $$a0a0a$$, now if a is between 9 and 100, let $$a =10a_1+a_0 \Rightarrow 10101*a=(10a_1*10^4+10a_1*10^2+1)+(a_0*10^4+a_0*10^2+a_0) = a_1*10^5+a_0*10^4+a_1*10^3+a_0*10^2+a_1*10+a_0=a_1a_0a_1a_0a_1a_0=aaa$$ in digit form.

So, it's no "coincidence" if you could even call it that. But this is basically the "reasons" for that it is what it is.

 

[arildno]

Well, you have already stated the relationship between 777 and 13: 777*13=10101

Now, multiplying 10101 with any two-digit number will, of course, result in a triple repetition of that number.

There's not very much to add to this; if you multiply 1010101 with a two-digit number, that number will be repeated four times, and so on.

 

[joeknockz]

Notice that $$777=7*111$$, so $$13*777=7*13*111=91*111=(91)(100+10+1)=9100+910+91$$ which you will se have a cancelling and shifting effect, as 9+1=10.

ahhhhhh I see it clearly now...the 13*7=91 was what was also confusing me earlier when 13*77=1001, 13*777=10101, 13*7777=101101
I see the reason clearly now...it's still kind of neat that those 2 numbers happen to be one lucky number and one unlucky number (both whole) that result in such a pattern..

 

[CompuChip]

It's just a coincidence that 777 x 13 = 10101.
That 777 x 13 x ab = ababab is then a trivial consequence -- just do the multiplication:

 10101
    ab x
--------
    ab
    00
  ab00
  0000
ab0000 +
--------
ababab

You can construct as many of these as you like, for example,
10001 * 101 = 1010101
and then e.g. 10001 * 101 * 65 = 65656565

217272749 * 4649 = 1010101010101
and then e.g. 217272749 * 4649 * 65 = 65656565656565

et cetera.

You can even make it work for bigger numbers, for example
170017 * 5882353
has the property that if you multiply it by a number n it will give n four times in a row, for any n between 1 and 9999:
5031 * 170017 * 5882353 = 5031503150315031

 

[arildno]

ahhhhhh I see it clearly now...the 13*7=91 was what was also confusing me earlier when 13*77=1001, 13*777=10101, 13*7777=101101
I see the reason clearly now...it's still kind of neat that those 2 numbers happen to be one lucky number and one unlucky number (both whole) that result in such a pattern..

Lucky and unlucky numers??

Man, do not confuse numerology with mathematics!

 

[joeknockz]

? just sayin i thought it was neat..

 

[CaptainQuasar]

It is neat, Joe. The branch of mathematics that would study this, BTW, (except for the luck part) is called http://mathworld.wolfram.com/NumberTheory.html" were also fascinated with it.

Ramanujan supposedly said “Every number is a personal friend to me.” The British mathematician G.H. Hardy told this story:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”​

(Ramanujan died very young, in his thirties. Unfortunately for the rest of us, who knows what he could have accomplished.)

⚛​