The Quest for 700: Weekly GMAT Challenge (Answer)

There are several ways to attack this problem. One way is to rewrite 1/510 as (1/5)10, and then convert to decimals:

1/510 = (1/5)10 = (0.2)10

Now rewrite 0.2 as 2 × 10-1:

(0.2)10 = (2 × 10-1)10 = 210 × 10-10

Multiplying an integer (such as 210) by 10-10 moves the decimal 10 places to the left. So the digit ten places to the right in the decimal expansion of 1/510 is the units digit of the original integer.

If you don’t know that 210 = 1,024 off the top of your head, you can either find it manually or simply examine the pattern in the units digits of the powers of 2. These units digits repeat themselves:

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

etc.

The cycle is 2, 4, 8, 6 repeating. After two cycles of four digits, we need two more digits in the cycle, so we arrive at 4.