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interests / rec.puzzles / For integers m,n ( 0 <= n < m ) How many positive integers are inexpressible as ( 3m + n ) ?

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* For integers m,n ( 0 <= n < m ) How many positivehenh...@gmail.com
`* Re: For integers m,n ( 0 <= n < m ) How many positive integers areEdward Murphy
 `* Re: For integers m,n ( 0 <= n < m ) How many positive integers arehenh...@gmail.com
  `- Re: For integers m,n ( 0 <= n < m ) How many positive integers areMike Terry

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For integers m,n ( 0 <= n < m ) How many positive integers are inexpressible as ( 3m + n ) ?

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Subject: For integers m,n ( 0 <= n < m ) How many positive
integers are inexpressible as ( 3m + n ) ?
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sat, 13 Aug 2022 19:56 UTC

------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.

For integers m,n such that ( 0 <= n < m )

How many positive integers are inexpressible as ( 3m + n ) ?

How many positive integers are inexpressible as ( 5m + n ) ?

How many positive integers are inexpressible as ( 10m + n ) ?

------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.

Re: For integers m,n ( 0 <= n < m ) How many positive integers are inexpressible as ( 3m + n ) ?

<tdc838$s0s$1@gioia.aioe.org>

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From: emurph...@zoho.com (Edward Murphy)
Newsgroups: rec.puzzles
Subject: Re: For integers m,n ( 0 <= n < m ) How many positive integers are
inexpressible as ( 3m + n ) ?
Date: Sun, 14 Aug 2022 18:38:43 -0700
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 by: Edward Murphy - Mon, 15 Aug 2022 01:38 UTC

On 8/13/2022 12:56 PM, henh...@gmail.com wrote:

> For integers m,n such that ( 0 <= n < m )
>
> How many positive integers are inexpressible as ( 3m + n ) ?
>
> How many positive integers are inexpressible as ( 5m + n ) ?
>
> How many positive integers are inexpressible as ( 10m + n ) ?

[spoiler space]

For 3m + n: Looking at the smallest possible values of m, and then the
smallest possible values of n for each value of m, we can express:

3 = 3(1) + 0

6 = 3(2) + 0
7 = 3(2) + 1

9 = 3(3) + 0
10 = 3(3) + 1
11 = 3(3) + 2

At this point, all higher numbers are also expressible:

x = 3p + q -> x + 3 = 3(p + 1) + q
x + 1 = 3r + s -> x + 4 = 3(r + 1) + s
x + 2 = 3t + u -> x + 5 = 3(t + 1) + u

Similarly, for 5m + n, the inexpressible numbers are

1, 2, 3, 4,
6, 7, 8, 9,
12, 13, 14,
18, 19,
24

and for 10m + n, the inexpressible numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9,
11, 12, 13, 14, 15, 16, 17, 18, 19,
22, 23, 24, 25, 26, 27, 28, 29,
33, 34, 35, 36, 37, 38, 39,
44, 45, 46, 47, 48, 49,
55, 56, 57, 58, 59,
66, 67, 68, 69,
77, 78, 79,
88, 89,
99

In general, for xm + n, the inexpressible numbers are the ones that
*are* expressible as xm + n for some integers 0 <= m <= n < x.

Re: For integers m,n ( 0 <= n < m ) How many positive integers are inexpressible as ( 3m + n ) ?

<ba01518f-cc89-4463-bce2-aa76d08a8c27n@googlegroups.com>

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Subject: Re: For integers m,n ( 0 <= n < m ) How many positive integers are
inexpressible as ( 3m + n ) ?
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 by: henh...@gmail.com - Mon, 15 Aug 2022 17:31 UTC

(4 animals) https://i.imgur.com/vSJTwZP.jpeg

On Sunday, August 14, 2022 at 6:38:51 PM UTC-7, Edward Murphy wrote:
> On 8/13/2022 12:56 PM, henh...@gmail.com wrote:
>
> > For integers m,n such that ( 0 <= n < m )
> >
> > How many positive integers are inexpressible as ( 3m + n ) ?
> >
> > How many positive integers are inexpressible as ( 5m + n ) ?
> >
> > How many positive integers are inexpressible as ( 10m + n ) ?

> [spoiler space]
>
i suppose... Some ppl can do this (in their heads) within 20 seconds.

Do we have this type of a problem (or (gap) shape)
in a real math problem, or real physics / engineering problem ?

Re: For integers m,n ( 0 <= n < m ) How many positive integers are inexpressible as ( 3m + n ) ?

<GOGcnbxqTsmwF2f_nZ2dnZfqnPrNnZ2d@brightview.co.uk>

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From: news.dea...@darjeeling.plus.com (Mike Terry)
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 by: Mike Terry - Mon, 15 Aug 2022 18:03 UTC

On 15/08/2022 18:31, henh...@gmail.com wrote:
>
> (4 animals) https://i.imgur.com/vSJTwZP.jpeg
>
>
> On Sunday, August 14, 2022 at 6:38:51 PM UTC-7, Edward Murphy wrote:
>> On 8/13/2022 12:56 PM, henh...@gmail.com wrote:
>>
>>> For integers m,n such that ( 0 <= n < m )
>>>
>>> How many positive integers are inexpressible as ( 3m + n ) ?
>>>
>>> How many positive integers are inexpressible as ( 5m + n ) ?
>>>
>>> How many positive integers are inexpressible as ( 10m + n ) ?
>
>> [spoiler space]
>>
>
> i suppose... Some ppl can do this (in their heads) within 20 seconds.
>
>
> Do we have this type of a problem (or (gap) shape)
> in a real math problem, or real physics / engineering problem ?
>

You may be interested in the coin problem:

<https://en.wikipedia.org/wiki/Coin_problem>

and/or the stamp problem:

<https://en.wikipedia.org/wiki/Postage_stamp_problem>

Regards,
Mike.

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