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Anonymous Tue 12 Apr 05:44:20 2022 No.14387649 View Reply Original Report

Quoted By: >>14387656 >>14387815 >>14387864 >>14387867 >>14387985 >>14389797 >>14390209

After adding a zero between two digits of a number (written in base 10) it becomes 9 times as large. Where was the zero inserted?

Anonymous

Anonymous Tue 12 Apr 2022 05:48:46 No.14387656 Report

Quoted By: >>14387822 >>14389797

>>14387649
45 × 9 = 405

Anonymous

Anonymous Tue 12 Apr 2022 08:19:53 No.14387815 Report

Quoted By: >>14387817 >>14387822 >>14388142 >>14388181

>>14387649
Let a be the first digit, b be the second.

9(10a+b) = 100a + b
90a + 9b = 100a + b
8b = 10a
b = (4/5)a
So a=5, b=4 works.

Anonymous

Anonymous Tue 12 Apr 2022 08:20:54 No.14387817 Report

Quoted By:

>>14387815
Oh I’m retarded. The zero was inserted between the two digits. Lmao

Anonymous

Anonymous Tue 12 Apr 2022 08:24:55 No.14387822 Report

Quoted By: >>14388153 >>14389797

>>14387656
>>14387815
what about the general case

Anonymous

Anonymous Tue 12 Apr 2022 08:33:57 No.14387834 Report

Quoted By: >>14387861 >>14388181 >>14389770

Any number of the form 9a s.t. a < 10^k and 10^k divides 8a for some k. Clearly, for any such number, you can add a zero to the end, so I will only list the "primitive ones", i.e. where a is not a multiple of 10. Those are 45, 225, 675, 1125, 3375, 5625 and 7875.

Anonymous

Anonymous Tue 12 Apr 2022 08:59:25 No.14387861 Report

Quoted By:

>>14387834
amazing

Anonymous

Anonymous Tue 12 Apr 2022 09:02:14 No.14387864 Report

Quoted By:

>>14387649
between the first and the second digit

Anonymous

Anonymous Tue 12 Apr 2022 09:06:40 No.14387867 Report

Quoted By:

>>14387649
>After adding a zero between two digits of a number
>Where was the zero inserted?

You're retarded.

Anonymous

Anonymous Tue 12 Apr 2022 10:46:08 No.14387985 Report

Quoted By:

>>14387649
>Where was the zero inserted?
In the middle
What do I win?

Anonymous

Anonymous Tue 12 Apr 2022 11:57:43 No.14388142 Report

Quoted By:

>>14387815
>8b = 10a
>b = (4/5)a
no,
a = (4/5)b

Anonymous

Anonymous Tue 12 Apr 2022 12:03:33 No.14388153 Report

Quoted By:

>>14387822

$9sum_{i=1}^n 10^i a_i = \sum_{i=1}^{m-1} 10^i a_i + \sum_{i=m+1}^{n+1} a_i$

Anonymous

Anonymous Tue 12 Apr 2022 12:20:13 No.14388181 Report

Quoted By:

>>14387815
>>14387834
What is this, number theory?

Anonymous

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Anonymous Tue 12 Apr 2022 21:43:50 No.14389770 Report

Quoted By: >>14390228

>>14387834
adding a zero to the end of a number multiplies it by 10, not 9

Anonymous

Anonymous Tue 12 Apr 2022 21:51:25 No.14389797 Report

Quoted By: >>14389807

>>14387822
>>14387649
>After adding a zero between two digits of a number (written in base 10) it becomes 9 times as large. Where was the zero inserted?
it doesn't matter as long as it's not the first or the last
if the sum of the digits of any number is divisible by 9, so will be the number itself

for example
>>14387656
40000000005 is divisible by 9, so is 800000000001

Anonymous

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Anonymous Tue 12 Apr 2022 21:55:08 No.14389807 Report

Quoted By: >>14389822

>>14389797

Anonymous

Anonymous Tue 12 Apr 2022 22:01:46 No.14389822 Report

Quoted By: >>14389991

>>14389807
I said it's divisible by 9 like op asks

Anonymous

Anonymous Tue 12 Apr 2022 22:53:28 No.14389991 Report

Quoted By:

>>14389822
https://educationusa.state.gov/your-5-steps-us-study/research-your-options/english-language/english-language-esl-programs

Anonymous

Anonymous Tue 12 Apr 2022 23:52:28 No.14390209 Report

Quoted By:

>>14387649

Suppose $a,b\in\mathbb{N}$ .
The length of $b$ being $l_b = \left \lfloor \log_{10} b\right \rfloor + 1$ .
Consider $\overline{ab} = 10^{l_b}a + b$ .
Adding $0$ at the start of $b$
$\overline{a0b} = 10\cdot 10^{l_b}a + b$
So now $\overline{a0b} = 9\overline{ab} \iff 10^{l_b}a = 8b \iff 10^{\left \lfloor \log_{10} b\right \rfloor + 1}a = 8b \iff 5^{\left \lfloor \log_{10}b \right \rfloor + 1}\cdot 2^{\left \lfloor \log_{10} b\right \rfloor - 2}a = b$ .

If I'm not mistaken the set of numbers such that the above condition is satisfied has all the numbers you seek.
I've written the condition like that to put emphasis on the divisibility rules that it implies.
A nice observation is that $a$ can't be more than 1 digit because $8b$ has no more digits than $l_b + 1$ .
That means that the 0 can only be inserted after the first digit.
\thread

Anonymous

Anonymous Wed 13 Apr 2022 00:00:37 No.14390228 Report

Quoted By:

>>14389770
you are retarded, how is that related

Anonymous

Anonymous Wed 13 Apr 2022 04:44:48 No.14390808 Report

Quoted By:

any integer can be written like xy, where x is the first series of digits and y is the last series of digits. so 12345 could be written like x=12300, y=45
well inserting a zero is like saying 10x + y. and multiplying by nine is like saying 9x+9y. so the equation for equality would be: x = 8y
this works for say 45, where y=5 and x=40. or y=25 and x=200. just as long as digits of x and y don't overlap

Anonymous

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Anonymous Wed 13 Apr 2022 05:13:50 No.14390840 Report

Quoted By:

bros...?

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