1987 AMC 12 Problem 16

Question

A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base

5.

Second, a

1

-to-

1

correspondence is established between the digits that appear in the expressions in base

5

and the elements of the set

\{V,W,X,Y,Z\}.

Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as

VYZ,

VYX,

VVW,

respectively. What is the base-

10

expression for the integer coded as

XYZ?

48

71

82

108

113

Compare the final digits of

VYZ

and

VYX

The change from

VYX

to

VVW

forces a base-

5

carry

Because

VYX

is one more than

VYZ,

the digit for

X

is one more than the digit for

Z.

The next increment changes

VYX

to

VVW,

so

X=4,\ W=0,

and

Z=3.

The carry changes

Y

to

V,

leaving

Y=1,\ V=2.

Thus

\begin{aligned} XYZ_5&=413_5,\\ 413_5&=4\cdot25+5+3\\ &=108. \end{aligned}

Thus the correct answer is D.

Problem 16 in other years