Inductive reasoning!

Prove that all numbers above 12 can be expressed in the form $7x + 3y$ by any method. Preferably by mathematical induction.

$x$ may be equal to $y$ and,
$x$ and $y$ are positive integers.

#Easy #Induction

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The answer is trivial by Postage Stamp Problem / Chicken McNugget Theorem.

Pi Han Goh - 5 years, 11 months ago

Yeah! I had thought of this. But I was not that sure to post it :)

Nihar Mahajan - 5 years, 11 months ago

Same here! BTW, did you think of anyway to prove it? If yes do share it.

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – I am not able to understand what exactly you want us to prove now?

Nihar Mahajan - 5 years, 11 months ago

@Nihar Mahajan – I want you to prove that all numbers above $12$ can be expressed in the form $7x + 3y$

-_-

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – Do you know what Chicken-Nugget theorem is? Isn't it has become obvious now? What do you think you need to prove now? @Sravanth Chebrolu

@Pi Han Goh Do we require more proof? It seems Chicken Nugget does all the work.

Oh Wait! Sravanth do you want us to prove Chicken Nugget theorem?

Nihar Mahajan - 5 years, 11 months ago

@Nihar Mahajan – No! Okay, you say it's become obvious that no such minimum number exists. Well then, why don't you tell me maximum the number which cannot be expressed as $3x+7y$ using Chicken Nugget theorem; I am not much familiar with it. :P

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – It is of course $11$ ! -_- (if x,y are non- negative)

Nihar Mahajan - 5 years, 11 months ago

@Nihar Mahajan – Then, it's by that theorem that we've proved that $3x+7y$ can be expressed by any number above $12$ isn't it?(Or, is it? maybe I'm mistaken)

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – Chicken-Mcnugget theorem states:

Let $m$ and $n$ be positive coprime integers. Then the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a$ and $b$ (called the Frobenius number) is $mn-m-n$ .

Here $m,n=(7,3)$ and substitution gives us answer as $11$ . This is a universally accepted theorem. If you have any problem with this , do tell me. It seems you didn't know this theorem at all.

Nihar Mahajan - 5 years, 11 months ago

@Nihar Mahajan – Not until today. Anyways, thanks for co-operating with me(of course thanks for teaching me a new theorem!) . I think I should leave now, bye!(I heard rumors that you're active on nights too! So, enjoy your learning. . . . :P)

Sravanth C. - 5 years, 11 months ago

Yes sir. I thought of it before posting this note. Actually the question was "You are in a world where, there are only denominations of $\$7$ and $\$3$ . What is the minimum number, above which you can obtain all possible values?"

So, what do you think sir? Can't there be an inductive proof for this?

Sravanth C. - 5 years, 11 months ago

"What is the minimum number, above which you can obtain all possible values?"

That is an impossible scenario isn't it?

Pi Han Goh - 5 years, 11 months ago

@Pi Han Goh – Yeah, I thought so sir. But taking a second look thought it's quite possible. When I tried by hit and trial I found that $12$ is such number, but I wasn't able to prove it. . . Can you please help me?

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – I've already given you the answer above.

Pi Han Goh - 5 years, 11 months ago

@Pi Han Goh – So, you mean it's impossible to prove it?

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – It's impossible to prove it because there is no maximum value. Please read the Wiki page.

Pi Han Goh - 5 years, 11 months ago

I think the most intuitive way to go about this is by expressing the $10$ numbers after $12$ (as a set, $S=\{13,14,15,...,22\}$ ). Once that is shown, the proof is essentially over, because we can just add $10$ to any number in $S$ , which will let us form any $n>12$ as $7x+3y$ , with $x$ and $y$ being non-negative integers.

Clearly, setting $x=0$ allows us to attain any multiple of $3$ , which in $S$ will give us $15,18,$ and $21$ . Similarly, we can set $y=0$ to attain any multiple of $7$ , which in $S$ will give us $14$ and $21$ . So, now we have the in-between cases of $x,y \neq 0$ . We can pluck the right $x$ and $y$ rather easily for these. $13 = 7(1)+3(2)$ $16= 7(1)+3(3)$ $17 = 7(2)+3(1)$ $19=7(1)+3(4)$ $20=7(2)+3(2)$ $22=7(1)+3(5)$ So, now we have a corresponding set of pairs $(x,y)$ which give us each number in $S$ , and we'll call this set $S*=\{(x_{13},y_{13}),(x_{14},y_{14}), ... , (x_{22},y_{22})\}$ .

Now, any $n>12$ can be formed. Say we want $26$ . Then we take the pair that gave us $16$ and we add one to both $x$ and $y$ : $7(1+1)+3(3+1)=26$ . So all we have done is add $10$ to our $16$ . If we want to add $40$ to an element of $S$ , we just add $4$ to each $x,y$ in the appropriate element of $S*$ . In general, we can add any multiple of $10$ by simply adding $n$ to each $x$ and $y$ of the appropriate pair. Thus we can express any $n>12$ as $7x+3y$ for positive $x$ and $y$ .

It's not the most elegant proof, but it works. I'd love to see a more elegant proof.

Ryan Tamburrino - 5 years, 11 months ago

Well done sir! But, this indeed doesn't prove that all numbers can be surely expressed as $3x+8y$ . Anyway it was a very good attempt sir!

Sravanth C. - 5 years, 11 months ago

What is it lacking?

Ryan Tamburrino - 5 years, 11 months ago

@Ryan Tamburrino – There is no proof that combining the results will give surely give out all the numbers. But, that is not a big issue sir, you've done very well. Can you try it using induction?

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – Ah! Chicken McNugget Theorem! How could I forget. There you go.

Ryan Tamburrino - 5 years, 11 months ago

Are $x$ and $y$ non-negative integers?

Ryan Tamburrino - 5 years, 11 months ago

Yeah. Thanks for asking! I've modified it. $\huge\ddot\smile$

Sravanth C. - 5 years, 11 months ago

If they are, $15$ is not possible. Do you mean that they are integers (positive or negative)?

A Former Brilliant Member - 5 years, 11 months ago

@A Former Brilliant Member – Why not? $15=7\times0+3\times5$

Sravanth C. - 5 years, 11 months ago

@Sravanth C. – Oh, sorry! I forgot about 0.

A Former Brilliant Member - 5 years, 11 months ago

@A Former Brilliant Member – Non-negative includes 0.

Ryan Tamburrino - 5 years, 11 months ago

@Sharky Kesa, @Daniel Liu, @Michael Mendrin sir, @Azhaghu Roopesh M sir can you post your own proofs?

Sravanth C. - 5 years, 11 months ago

@Calvin Lin sir and @Nihar Mahajan. Can you post your proofs? Please. . .

Sravanth C. - 5 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$