What is the digit of the $1997^{\mathrm{th}}$ decimal place of:
(a) $\frac{1}{22}$
(b) $\frac{1}{27}$

## Level 1 Solutions

## List 1

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(a) Dividing 1 by 22 we get: $\frac{1}{22}=0.0454545 \ldots$ We observe that the digit 4 is in the even positions: $2,4,6, \ldots$ and the digit 5 in the odd positions: $3,5,7 \ldots$ Since 1997 is an odd number, the digit in the $1997^{\text{th}}$ decimal place is 5.

(b) Dividing 1 by 27 we get: $\frac{1}{27}=0.037037037 \ldots$

We observe that the digits 0, 3, and 7 repeat, successively, every three decimal places, with the digit:

- 0 in the positions: $1,4,7, \ldots$, that is, when divided by three, they leave a remainder of 1;
- 3 in the positions: $2,5,8, \ldots$, that is, when divided by three, they leave a remainder of 2;
- 7 in the positions: $3,6,9, \ldots$, that is, they are multiples of 3.

Since the division $1997 \div 3$ leaves a remainder of 2, the digit in the $1997^{\text{th}}$ decimal place is 3.

![](https://cdn.mathpix.com/cropped/2024_05_01_b9d1398cc05f7e479435g-6.jpg?height=119&width=209&top_left_y=2459&top_left_x=1506)