Find all 2-digit

Let us start by collecting all 2-digit numbers.

They are from 10 – 99.

From these numbers, we have prime numbers also which are not divisible by any number except 1.

They are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

For this, we need to check every 2-digit number for its divisibility by the sum of its digits.

For 10,

The digits are 1 and 0.

Sum of digits = 1 + 0 = 1

And 10 is divisible by 1 obviously.

⇒ 10 is divisible by the sum of its digits.

For 12,

The digits are 1 and 2.

Sum of digits = 1 + 2 = 3

And 12 is divisible by 3.

⇒ 12 is divisible by the sum of its digits.

For 14,

The digits are 1 and 4.

Sum of digits = 1 + 4 = 5

And 14 is not divisible by 5.

⇒ 14 is not divisible by the sum of its digits.

For 15,

The digits are 1 and 5.

Sum of digits = 1 + 5 = 6

And 15 is not divisible by 6.

⇒ 15 is not divisible by the sum of its digits.

For 16,

The digits are 1 and 6.

Sum of digits = 1 + 6 = 7

And 16 is not divisible by 7.

⇒ 16 is not divisible by the sum of its digits.

For 18,

The digits are 1 and 8.

Sum of digits = 1 + 8 = 9

And 18 is divisible by 9.

⇒ 18 is divisible by the sum of its digits.

For 20,

The digits are 2 and 0.

Sum of digits = 2 + 0 = 2

And 20 is divisible by 2.

⇒ 20 is divisible by the sum of its digits.

For 21,

The digits are 2 and 1.

Sum of digits = 2 + 1 = 3

And 21 is divisible by 3.

⇒ 21 is divisible by the sum of its digits.

For 22,

The digits are 2 and 2.

Sum of digits = 2 + 2 = 4

And 22 is not divisible by 4.

⇒ 22 is not divisible by the sum of its digits.

For 24,

The digits are 2 and 4.

Sum of digits = 2 + 4 = 6

And 24 is divisible by 6.

⇒ 24 is divisible by the sum of its digits.

For 25,

The digits are 2 and 5.

Sum of digits = 2 + 5 = 7

And 25 is not divisible by 7.

⇒ 25 is not divisible by the sum of its digits.

For 26,

The digits are 2 and 6.

Sum of digits = 2 + 6 = 8

And 26 is not divisible by 8.

⇒ 26 is not divisible by the sum of its digits.

For 27,

The digits are 2 and 7.

Sum of digits = 2 + 7 = 9

And 27 is divisible by 9.

⇒ 27 is divisible by the sum of its digits.

For 28,

The digits are 2 and 8.

Sum of digits = 2 + 8 = 10

And 28 is not divisible by 10.

⇒ 28 is not divisible by the sum of its digits.

For 30,

The digits are 3 and 0.

Sum of digits = 3 + 0 = 3

And 30 is divisible by 3.

⇒ 30 is divisible by the sum of its digits.

For 32,

The digits are 3 and 2.

Sum of digits = 3 + 2 = 5

And 32 is not divisible by 5.

⇒ 32 is not divisible by the sum of its digits.

For 33,

The digits are 3 and 3.

Sum of digits = 3 + 3 = 6

And 33 is not divisible by 6.

⇒ 33 is not divisible by the sum of its digits.

For 34,

The digits are 3 and 4.

Sum of digits = 3 + 4 = 7

And 34 is not divisible by 7.

⇒ 34 is not divisible by the sum of its digits.

For 35,

The digits are 3 and 5.

Sum of digits = 3 + 5 = 8

And 35 is not divisible by 8.

⇒ 35 is not divisible by the sum of its digits.

For 36,

The digits are 3 and 6.

Sum of digits = 3 + 6 = 9

And 36 is divisible by 9.

⇒ 36 is divisible by the sum of its digits.

For 38,

The digits are 3 and 8.

Sum of digits = 3 + 8 = 11

And 38 is not divisible by 11.

⇒ 38 is not divisible by the sum of its digits.

For 39,

The digits are 3 and 9.

Sum of digits = 3 + 9 = 12

And 39 is not divisible by 12.

⇒ 39 is not divisible by the sum of its digits.

For 40,

The digits are 4 and 0.

Sum of digits = 4 + 0 = 4

And 40 is divisible by 4.

⇒ 40 is divisible by the sum of its digits.

For 42,

The digits are 4 and 2.

Sum of digits = 4 + 2 = 6

And 42 is divisible by 6.

⇒ 42 is divisible by the sum of its digits.

For 44,

The digits are 4 and 4.

Sum of digits = 4 + 4 = 8

And 44 is not divisible by 8.

⇒ 44 is not divisible by the sum of its digits.

For 45,

The digits are 4 and 5.

Sum of digits = 4 + 5 = 9

And 45 is divisible by 9.

⇒ 45 is divisible by the sum of its digits.

For 46,

The digits are 4 and 6.

Sum of digits = 4 + 6 = 10

And 46 is not divisible by 10.

⇒ 46 is not divisible by the sum of its digits.

For 48,

The digits are 4 and 8.

Sum of digits = 4 + 8 = 12

And 48 is divisible by 12.

⇒ 48 is divisible by the sum of its digits.

For 49,

The digits are 4 and 9.

Sum of digits = 4 + 9 = 13

And 49 is not divisible by 13.

⇒ 49 is not divisible by the sum of its digits.

For 50,

The digits are 5 and 0.

Sum of digits = 5 + 0 = 5

And 50 is divisible by 5.

⇒ 50 is divisible by the sum of its digits.

For 51,

The digits are 5 and 1.

Sum of digits = 5 + 1 = 6

And 51 is not divisible by 6.

⇒ 51 is not divisible by the sum of its digits.

For 52,

The digits are 5 and 1.

Sum of digits = 5 + 1 = 6

And 51 is not divisible by 6.

⇒ 51 is not divisible by the sum of its digits.

For 54,

The digits are 5 and 4.

Sum of digits = 5 + 4 = 9

And 54 is divisible by 9.

⇒ 54 is divisible by the sum of its digits.

For 55,

The digits are 5 and 5.

Sum of digits = 5 + 5 = 10

And 55 is not divisible by 10.

⇒ 55 is not divisible by the sum of its digits.

For 56,

The digits are 5 and 6.

Sum of digits = 5 + 6 = 11

And 56 is not divisible by 11.

⇒ 56 is not divisible by the sum of its digits.

For 57,

The digits are 5 and 7.

Sum of digits = 5 + 7 = 12

And 57 is not divisible by 12.

⇒ 57 is not divisible by the sum of its digits.

For 58,

The digits are 5 and 8.

Sum of digits = 5 + 8 = 13

And 58 is not divisible by 13.

⇒ 58 is not divisible by the sum of its digits.

For 60,

The digits are 6 and 0.

Sum of digits = 6 + 0 = 6

And 60 is divisible by 6.

⇒ 60 is divisible by the sum of its digits.

For 62,

The digits are 6 and 2.

Sum of digits = 6 + 2 = 8

And 62 is not divisible by 8.

⇒ 62 is not divisible by the sum of its digits.

For 63,

The digits are 6 and 3.

Sum of digits = 6 + 3 = 9

And 63 is divisible by 9.

⇒ 63 is divisible by the sum of its digits.

For 64,

The digits are 6 and 4.

Sum of digits = 6 + 4 = 10

And 64 is not divisible by 10.

⇒ 64 is not divisible by the sum of its digits.

For 65,

The digits are 6 and 5.

Sum of digits = 6 + 5 = 11

And 65 is not divisible by 11.

⇒ 65 is not divisible by the sum of its digits.

For 66,

The digits are 6 and 6.

Sum of digits = 6 + 6 = 12

And 66 is not divisible by 12.

⇒ 66 is not divisible by the sum of its digits.

For 70,

The digits are 7 and 0.

Sum of digits = 7 + 0 = 7

And 70 is divisible by 7.

⇒ 70 is divisible by the sum of its digits.

For 72,

The digits are 7 and 2.

Sum of digits = 7 + 2 = 9

And 72 is divisible by 9.

⇒ 72 is divisible by the sum of its digits.

For 74,

The digits are 7 and 4.

Sum of digits = 7 + 4 = 11

And 74 is not divisible by 11.

⇒ 74 is not divisible by the sum of its digits.

For 75,

The digits are 7 and 5.

Sum of digits = 7 + 5 = 12

And 75 is not divisible by 12.

⇒ 75 is not divisible by the sum of its digits.

For 76,

The digits are 7 and 6.

Sum of digits = 7 + 6 = 13

And 76 is not divisible by 13.

⇒ 76 is not divisible by the sum of its digits.

For 77,

The digits are 7 and 7.

Sum of digits = 7 + 7 = 14

And 77 is not divisible by 14.

⇒ 77 is not divisible by the sum of its digits.

For 78,

The digits are 7 and 8.

Sum of digits = 7 + 8 = 15

And 78 is not divisible by 15.

⇒ 78 is not divisible by the sum of its digits.

For 80,

The digits are 8 and 0.

Sum of digits = 8 + 0 = 8

And 80 is divisible by 8.

⇒ 80 is divisible by the sum of its digits.

For 81,

The digits are 8 and 1.

Sum of digits = 8 + 1 = 9

And 81 is divisible by 9.

⇒ 81 is divisible by the sum of its digits.

For 82,

The digits are 8 and 2.

Sum of digits = 8 + 2 = 10

And 82 is not divisible by 10.

⇒ 82 is not divisible by the sum of its digits.

For 84,

The digits are 8 and 4.

Sum of digits = 8 + 4 = 12

And 84 is divisible by 12.

⇒ 84 is divisible by the sum of its digits.

For 85,

The digits are 8 and 5.

Sum of digits = 8 + 5 = 13

And 85 is not divisible by 13.

⇒ 85 is not divisible by the sum of its digits.

For 86,

The digits are 8 and 6.

Sum of digits = 8 + 6 = 14

And 86 is not divisible by 14.

⇒ 86 is not divisible by the sum of its digits.

For 87,

The digits are 8 and 7.

Sum of digits = 8 + 7 = 15

And 87 is not divisible by 15.

⇒ 87 is not divisible by the sum of its digits.

For 88,

The digits are 8 and 8.

Sum of digits = 8 + 8 = 16

And 88 is not divisible by 16.

⇒ 88 is not divisible by the sum of its digits.

For 90,

The digits are 9 and 0.

Sum of digits = 9 + 0 = 9

And 90 is divisible by 9.

⇒ 90 is divisible by the sum of its digits.

For 91,

The digits are 9 and 1.

Sum of digits = 9 + 1 = 10

And 91 is not divisible by 10.

⇒ 91 is not divisible by the sum of its digits.

For 92,

The digits are 9 and 2.

Sum of digits = 9 + 2 = 11

And 92 is not divisible by 11.

⇒ 92 is not divisible by the sum of its digits.

For 93,

The digits are 9 and 3.

Sum of digits = 9 + 3 = 12

And 93 is not divisible by 12.

⇒ 93 is not divisible by the sum of its digits.

For 94,

The digits are 9 and 4.

Sum of digits = 9 + 4 = 13

And 94 is not divisible by 13.

⇒ 94 is not divisible by the sum of its digits.

For 95,

The digits are 9 and 5.

Sum of digits = 9 + 5 = 14

And 95 is not divisible by 14.

⇒ 95 is not divisible by the sum of its digits.

For 96,

The digits are 9 and 6.

Sum of digits = 9 + 6 = 15

And 96 is not divisible by 15.

⇒ 96 is not divisible by the sum of its digits.

For 98,

The digits are 9 and 8.

Sum of digits = 9 + 8 = 17

And 98 is not divisible by 17.

⇒ 98 is not divisible by the sum of its digits.

For 99,

The digits are 9 and 2.

Sum of digits = 9 + 2 = 11

And 92 is not divisible by 11.

⇒ 92 is not divisible by the sum of its digits.

Thus, the answer is 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90.