All of the following relationships are correct, except one. Determine the wrong answer and explain why the others are correct.

A) 5612 ~ 1475
B) 4657 ~ 1210
C) 4326 ~ 1080
D) 2474 ~ 755
E) 1781 ~ 737

The problem's title, "Column Bases", hints at the relationship between the left and right columns of numbers.

The numbers of the left column have a "missing" mixed radix.  For each bⁿ positional digit ("column") of each number in left column of numbers,  the "missing" radix for the position is the minimal base the digits in that position can be represented in, i.e., the radix should be n+1 where n is the largest digit in the position.  When converted to base 10, the number in the left column,  except the wrong answer, equals the number in the right column on the same line.

For each digit in the b³ position (thousands digit in base 10) of the left column, the base would be 6 as the largest digit is 5.
For each digit in the b² position (hundreds digit in base 10) of the left column, the base would be 8 as the largest digit is 7.
For each digit in the b¹ position (tens digit in base 10) of the left column, the base would be 9 as the largest digit is 8.
For each digit in the b⁰ position (ones digit in base 10) of the left column, the base would be 8 as the largest digit is 7.

A) 5612 ~ 5₆6₈1₉2₈ = 1475
B) 4657 ~ 4₆6₈5₉7₈ = 1300  <-- right answer (not 1210)
C) 4326 ~ 4₆3₈2₉6₈ = 1080
D) 2474 ~ 2₆4₈7₉4₈ =  755
E) 1781 ~ 1₆7₈8₉1₈ =  737

As can be seen, B is the answer as it is the only item that does not follow this relationship, i.e., 1210 is not equal 1300.

Edited on February 27, 2008, 12:50 pm

Posted by Dej Mar on 2008-02-26 11:17:34