Jonas wrote a sequence with the positive multiples of 13 in ascending order.

## $1326395265 \ldots$

a) What is the $2,019^{\text{th}}$ digit of Jonas's sequence?

b) Will the number 2,019 appear in this sequence?
Solution

a) There are 7 multiples of 13 with 2 digits (14 digits); with 3 digits, there are 69 multiples of $13(3 \cdot 69=207$ digits). Already, there are $14+207=221$ digits, so $2,019-221=1,798$ are still needed. Dividing 1,798 by 4, we get 449 and a remainder of 2, that is, the first multiple of 13 with 4 digits is $13 \cdot 77=1001$, so $13(449+76)=6,825$. Since the remainder of the division is 2 and the next multiple is 6,838, then the $2,019^{\text{th}}$ digit is 8.

b) Yes. Dividing 20,190 by 13, we find the quotient is 1,553 and the remainder is 1, so 20,189 and 20,202 are multiples of 13, which means there is no multiple of 13 between 20,190 and 20,199. But, dividing 201,900 by 13, we find the quotient is 15,530 and the remainder is 10, so 201,903 is a multiple of 13 and, consequently, 2,019 appears in the sequence.