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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