• #### For integers m,n ( 0 <= n < m ) How many positive integers are inexpres

From henhanna@gmail.com@21:1/5 to All on Sat Aug 13 12:56:55 2022

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 ) ?

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• From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Aug 14 18:38:43 2022
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.

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• From henhanna@gmail.com@21:1/5 to Edward Murphy on Mon Aug 15 10:31:29 2022
(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 ?

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• From Mike Terry@21:1/5 to henh...@gmail.com on Mon Aug 15 19:03:07 2022
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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