Problem of the Week 16

What is the least positive integer meeting each of the following conditions?

Dividing by 7 gives a remainder of 4.
Dividing by 8 gives a remainder of 5.
Dividing by 9 gives a remainder of 6.

 

Alternate Problem 16

The base 10 (decimal) number system has 10 different digits.  The base 16 number system has 16 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F, where the base 10 equivalents of A, B, C, D, E, and F are 10, 11, 12, 13, 14, and 15, respectively.  How is 8,794,361 written in base 16?

 

Extension Problem 16

The place values in a base 3 number system are powers of 3, namely 1, 3, 9, 27, 81, and so on.  Suppose the digits in a base 3 number system are 1, 0, and 1, where 1 means one, 0 means zero, and 1 means negative one. The base 10 number 35 is written as 1101 in this base system.

a.  Write the base 3 notations for the base 10 numbers 1 through 10.

b.  Show how to multiply 101111 by 11101 in this base 3 number system.