A palindrome is a positive integer which reads the same forward and backwards, like 12321 or 8448.

The digits have to add up to a multiple of 3

The biggest one will be 9999 

Lets see

There are loads of patterns happening here - can you find them all ??      

1221             2

1551             5

1881             8

2112             1

2442             4

2772             7

3003             0

3333             3

3663             6

3993             9

4224             2

4554

4884

5115             1

5445

5775

6006              0

6336             

6666

6996 

7227             2

7557

7887 

8118              1

8448

8778

9009              0

9339

9669

9999

6*3 + 3*4 = 18+12 = 30

There are 30 four digit  palindromes  

Proper numbers only, I didn't include any begining with 0 but if i had there would be 4 or those.

A palindrome is a positive integer which reads the same forward and backwards, like  12321 or 8448.
How many 4-digit palindromes are divisible by 3?

\(\begin{array}{|r|r|r|r|} \hline & n & \text{palindrome } a(n) & \text{divisible by } 3 \\ \hline & 110& 10\ 01 \\ & 111& 11\ 11 \\ 1. & 112& 12\ 21 & \checkmark \\ & 113& 13\ 31 \\ & 114& 14\ 41 \\ 2. & 115& 15\ 51 &\checkmark \\ & 116& 16\ 61 \\ & 117& 17\ 71 \\ 3. & 118& 18\ 81 &\checkmark \\ & 119& 19\ 91 \\ \hline & 120& 20\ 02 \\ 4. & 121& 21\ 12 &\checkmark \\ & 122& 22\ 22 \\ & 123& 23\ 32 \\ 5. & 124& 24\ 42 &\checkmark \\ & 125& 25\ 52 \\ & 126& 26\ 62 \\ 6. & 127& 27\ 72 &\checkmark \\ & 128& 28\ 82 \\ & 129& 29\ 92 \\ \hline 7. & 130& 30\ 03 &\checkmark \\ & 131& 31\ 13 \\ & 132& 32\ 23 \\ 8. & 133& 33\ 33 &\checkmark \\ & 134& 34\ 43 \\ & 135& 35\ 53 \\ 9. & 136& 36\ 63 &\checkmark \\ & 137 & 37\ 73 \\ & 138& 38\ 83 \\ 10. & 139& 39\ 93 &\checkmark \\ \hline & 140& 40\ 04 \\ & 141& 41\ 14 \\ 11. & 142& 42\ 24 &\checkmark \\ & 143& 43\ 34 \\ & 144& 44\ 44 \\ 12. & 145& 45\ 54 &\checkmark \\ & 146& 46\ 64 \\ & 147& 47\ 74 \\ 13. & 148& 48\ 84 &\checkmark \\ & 149& 49\ 94 \\ \hline & 150& 50\ 05 \\ 14. & 151& 51\ 15 &\checkmark \\ & 152& 52\ 25 \\ & 153& 53\ 35 \\ 15. & 154& 54\ 45 &\checkmark \\ & 155& 55\ 55 \\ & 156& 56\ 65 \\ 16. & 157& 57\ 75 &\checkmark \\ & 158& 58\ 85 \\ & 159& 59\ 95 \\ \hline 17. & 160& 60\ 06 &\checkmark \\ & 161& 61\ 16 \\ & 162& 62\ 26 \\ 18. & 163& 63\ 36 &\checkmark \\ & 164& 64\ 46 \\ & 165& 65\ 56 \\ 19. & 166& 66\ 66 &\checkmark \\ & 167& 67\ 76 \\ & 168& 68\ 86 \\ 20. & 169& 69\ 96 &\checkmark \\ \hline & 170& 70\ 07 \\ & 171& 71\ 17 \\ 21. & 172& 72\ 27 &\checkmark \\ & 173& 73\ 37 \\ & 174& 74\ 47 \\ 22. & 175& 75\ 57 &\checkmark \\ & 176& 76\ 67 \\ & 177& 77\ 77 \\ 23. & 178& 78\ 87 &\checkmark \\ & 179& 79\ 97 \\ \hline & 180& 80\ 08 \\ 24. & 181& 81\ 18 &\checkmark \\ & 182& 82\ 28 \\ & 183& 83\ 38 \\ 25. & 184& 84\ 48 &\checkmark \\ & 185& 85\ 58 \\ & 186& 86\ 68 \\ 26. & 187& 87\ 78 &\checkmark \\ & 188& 88\ 88 \\ & 189& 89\ 98 \\ \hline 27. & 190& 90\ 09 &\checkmark \\ & 191& 91\ 19 \\ & 192& 92\ 29 \\ 28. & 193& 93\ 39 &\checkmark \\ & 194& 94\ 49 \\ & 195& 95\ 59 \\ 29. & 196& 96\ 69 &\checkmark \\ & 197& 97\ 79 \\ & 198& 98\ 89 \\ 30. & 199& 99\ 99 &\checkmark \\ \hline \end{array}\)

30  4-digit palindromes are divisible by 3.