Math Is Fun Forum

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Such a number cannot be one digit long; it must have at least two digits. The number of such two-digits numbers is 9 (namely 19, 28, 37, 46, 55, 64, 73, 82, 91).

For the three-digit numbers, I group them into nine groups, according to the starting number:

Group 1: 109, 118, 127, 136, 145, 154, 163, 172, 181, 190
Group 2: 208, 217, 226, 235, 244, 253, 262, 271, 280
Group 3: 307, 316, 325, 334, 343, 352, 361, 370
Group 4: 406, 415, 424, 433, 442, 451, 460
Group 5: 505, 514, 523, 532, 541, 550
Group 6: 604, 613, 622, 631, 640
Group 7: 703, 712, 721, 730
Group 8: 802, 811, 820
Group 9: 901, 910

So there are 10 group-1 numbers, 9 group-2 numbers, etc. Total number of three-digit numbers: 54.

Now for the four-digit numbers. The number of four-digit numbers ending __00 is clearly the same as the number of two-digit numbers, namely 9.

For the four-digit numbers not ending __00, look at the first two digits. The number of four-digit numbers whose first two digits add up to n is the same as the number of group-n numbers. For example, the number of four-digit numbers beginning 72__ is the same as the number of group-9 numbers, the number of four-digit numbers beginning 34__ is the same as the number group-7 numbers, etc.

Group 1: 10__
Group 2: 11__, 20__
Group 3: 12__, 21__, 30__
Group 4: 13__, 22__, 31__, 40__
Group 5: 14__, 23__, 32__, 41__, 50__
Group 6: 15__, 24__, 33__, 42__, 51__, 60__
Group 7: 16__, 25__, 34__, 43__, 52__, 61__, 70__
Group 8: 17__, 26__, 35__, 44__, 53__, 62__, 71__, 80__
Group 9: 18__, 27__, 36__, 45__, 54__, 63__, 72__, 81__, 90__

Therefore there are 1×10 + 2×9 + 3×8 + 4×7 + 5×6 + 6×5 + 7×4 + 8×3 + 9×2 = 210 four-digit numbers not ending __00.

Hence the total number of natual numbers less than 10000 with digits adding up to 10 is 9 + 54 + 9 + 210 = 282.

1 - loud

2 - unemployment

3 - germs

Let the radius of the cylinder be r. Then its height is a(b−r)/b (by similar triangles). So

There's a rather cute name for this test. It's called the Plain Vanilla Comparison Test.