Problem: In the base ten number system the number 526 means 5⋅102+2⋅10+6. In the Land of Mathesis, however, numbers are written in the base r. Jones purchases an automobile there for 440 monetary units (abbreviated m.u.). He gives the salesman a 1000 m.u. bill, and receives, in change, 340 m.u. The base r is:
Answer Choices:
A. 2
B. 5
C. 7
D. 8
E. 12
Solution:
We are given that 1000 m.u,−340 m.u. =440 m.u. (base r)
∴1⋅r3+0⋅r2+0⋅r+0−(3r2+4r+0)
=4r2+4r+0∴r3−7r2−8r=0
∴r2−7r−8=0∴r=8
or
More simply, we have 440+340=1000. Since 4+4
terminates in 0 in the system being used, the system has modulus 8, that is, r=8.