In the numeration system with base 5, counting is as follows: 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, … The number whose description in the decimal system is 69, when described in the base 5 system, is a number with:

(D) three non-consecutive digits

1960 AMC 12 Problem 16

Attempt Problem 16 of the 1960 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1960 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

16.

In the numeration system with base

5,

counting is as follows:

1,

2,

3,

4,

10,

11,

12,

13,

14,

20,

\ldots

The number whose description in the decimal system is

69,

when described in the base

5

system, is a number with:

two consecutive digits

two non-consecutive digits

three consecutive digits

three non-consecutive digits

four digits

Express

69

using powers of

5

After finding the base-

5

digits, compare their values

Since

69=2\cdot25+3\cdot5+4,

its base-

5

representation is

234_5.

Its three digits

2,

3,

and

4

are consecutive.

Therefore, the correct answer is C.