A two-digit integer is written next to itself twice, forming a six-digit number. If the resulting number is divisible by 6, then how many possibilities are there for the original two-digit number?
+26367 A two-digit integer is written next to itself twice, forming a six-digit number.
If the resulting number is divisible by 6,
then how many possibilities are there for the original two-digit number?
The two-digit integer: ab
The six-digit number: ababab
\(\text{Divisible by $6$: If it is divisible by $2$ and by $3$ } \\ \text{Divisible by $2$: So $b$ must be even! $b=\{0,2,4,6,8\} $} \\ \text{Divisible by $3$: Sum the digits. The result must be divisible by $3$,} \\ \text{ $\qquad a+b+a+b+a+b = 3a+3b$, so ababab is always divisible by $3$ } \)
\(\begin{array}{|l|r|r|r|r|} \hline &a & b & ab & ababab \\ \hline 1.& 1 & 0 & 10 & 101010 \\ 2.& 1 & 2 & 12 & 121212 \\ 3.& 1 & 4 & 14 & 141414 \\ 4.& 1 & 6 & 16 & 161616 \\ 5.& 1 & 8 & 18 & 181818 \\ \hline 6.& 2 & 0 & 20 & 202020 \\ 7.& 2 & 2 & 22 & 222222 \\ 8.& 2 & 4 & 24 & 242424 \\ 9.& 2 & 6 & 26 & 262626 \\ 10.& 2 & 8 & 28 & 282828 \\ \hline 11.& 3 & 0 & 30 & 303030 \\ 12.& 3 & 2 & 32 & 323232 \\ 13.& 3 & 4 & 34 & 343434 \\ 14.& 3 & 6 & 36 & 363636 \\ 15.& 3 & 8 & 38 & 383838 \\ \hline 16.& 4 & 0 & 40 & 404040 \\ 17.& 4 & 2 & 42 & 424242 \\ 18.& 4 & 4 & 44 & 444444 \\ 19.& 4 & 6 & 46 & 464646 \\ 20.& 4 & 8 & 48 & 484848 \\ \hline 21.& 5 & 0 & 50 & 505050 \\ 22.& 5 & 2 & 52 & 525252 \\ 23.& 5 & 4 & 54 & 545454 \\ 24.& 5 & 6 & 56 & 565656 \\ 25.& 5 & 8 & 58 & 585858 \\ \hline 26.& 6 & 0 & 60 & 606060 \\ 27.& 6 & 2 & 62 & 626262 \\ 28.& 6 & 4 & 64 & 646464 \\ 29.& 6 & 6 & 66 & 666666 \\ 30.& 6 & 8 & 68 & 686868 \\ \hline 31.& 7 & 0 & 70 & 707070 \\ 32.& 7 & 2 & 72 & 727272 \\ 33.& 7 & 4 & 74 & 747474 \\ 34.& 7 & 6 & 76 & 767676 \\ 35.& 7 & 8 & 78 & 787878 \\ \hline 36.& 8 & 0 & 80 & 808080 \\ 37.& 8 & 2 & 82 & 828282 \\ 38.& 8 & 4 & 84 & 848484 \\ 39.& 8 & 6 & 85 & 868686 \\ 40.& 8 & 8 & 88 & 888888 \\ \hline 41.& 9 & 0 & 90 & 909090 \\ 42.& 9 & 2 & 92 & 929292 \\ 43.& 9 & 4 & 94 & 949494 \\ 44.& 9 & 6 & 96 & 969696 \\ 45.& 9 & 8 & 98 & 989898 \\ \hline \end{array}\)
There are 45 possibilities for the original two-digit number.
+26367 A two-digit integer is written next to itself twice, forming a six-digit number.
If the resulting number is divisible by 6,
then how many possibilities are there for the original two-digit number?
The two-digit integer: ab
The six-digit number: ababab
\(\text{Divisible by $6$: If it is divisible by $2$ and by $3$ } \\ \text{Divisible by $2$: So $b$ must be even! $b=\{0,2,4,6,8\} $} \\ \text{Divisible by $3$: Sum the digits. The result must be divisible by $3$,} \\ \text{ $\qquad a+b+a+b+a+b = 3a+3b$, so ababab is always divisible by $3$ } \)
\(\begin{array}{|l|r|r|r|r|} \hline &a & b & ab & ababab \\ \hline 1.& 1 & 0 & 10 & 101010 \\ 2.& 1 & 2 & 12 & 121212 \\ 3.& 1 & 4 & 14 & 141414 \\ 4.& 1 & 6 & 16 & 161616 \\ 5.& 1 & 8 & 18 & 181818 \\ \hline 6.& 2 & 0 & 20 & 202020 \\ 7.& 2 & 2 & 22 & 222222 \\ 8.& 2 & 4 & 24 & 242424 \\ 9.& 2 & 6 & 26 & 262626 \\ 10.& 2 & 8 & 28 & 282828 \\ \hline 11.& 3 & 0 & 30 & 303030 \\ 12.& 3 & 2 & 32 & 323232 \\ 13.& 3 & 4 & 34 & 343434 \\ 14.& 3 & 6 & 36 & 363636 \\ 15.& 3 & 8 & 38 & 383838 \\ \hline 16.& 4 & 0 & 40 & 404040 \\ 17.& 4 & 2 & 42 & 424242 \\ 18.& 4 & 4 & 44 & 444444 \\ 19.& 4 & 6 & 46 & 464646 \\ 20.& 4 & 8 & 48 & 484848 \\ \hline 21.& 5 & 0 & 50 & 505050 \\ 22.& 5 & 2 & 52 & 525252 \\ 23.& 5 & 4 & 54 & 545454 \\ 24.& 5 & 6 & 56 & 565656 \\ 25.& 5 & 8 & 58 & 585858 \\ \hline 26.& 6 & 0 & 60 & 606060 \\ 27.& 6 & 2 & 62 & 626262 \\ 28.& 6 & 4 & 64 & 646464 \\ 29.& 6 & 6 & 66 & 666666 \\ 30.& 6 & 8 & 68 & 686868 \\ \hline 31.& 7 & 0 & 70 & 707070 \\ 32.& 7 & 2 & 72 & 727272 \\ 33.& 7 & 4 & 74 & 747474 \\ 34.& 7 & 6 & 76 & 767676 \\ 35.& 7 & 8 & 78 & 787878 \\ \hline 36.& 8 & 0 & 80 & 808080 \\ 37.& 8 & 2 & 82 & 828282 \\ 38.& 8 & 4 & 84 & 848484 \\ 39.& 8 & 6 & 85 & 868686 \\ 40.& 8 & 8 & 88 & 888888 \\ \hline 41.& 9 & 0 & 90 & 909090 \\ 42.& 9 & 2 & 92 & 929292 \\ 43.& 9 & 4 & 94 & 949494 \\ 44.& 9 & 6 & 96 & 969696 \\ 45.& 9 & 8 & 98 & 989898 \\ \hline \end{array}\)
There are 45 possibilities for the original two-digit number.
