Question

if $aaa 4$ can be expressed as $33 b$, where $a$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $a b$?

Answer 1

if $aaa 4$ can be expressed as $33 b$, where $a$ is a digit in base 4 and $b$ is a base greater than 5, the smallest possible sum $a b$ is 7.

The number of digits or combinations of digits used in a counting system to represent numbers is called a number base. A base can be any entire number more prominent than 0. The decimal system, more commonly referred to as base 10, is the most widely used system of numbers.

In base four, the number of copies of a power of four is represented by each digit of the number. That is, the number of ones you have is indicated by the first digit; The second displays your total number of fours; The third section reveals your total number of sixteens or four-times-fours; The fourth part tells you how many sixty-fours you have, or four times four times four; and so forth.

$AAA_{4} = 33b$

Let A= 0, 1, 2, 3

$A.4^{2}+A.4^{1} +A.4^{0} = 3b^{1} +3b^{0} \\A.16+A.4+A=3b+3\\21A= 3. (b+1)\\7A=b+1\\b=7A-1$

A   b=7A-1   A+B (b>7)

0          -1        -1

1           16        7

2          13       15

3          20      23

$111_{4} = 33_{6}$

The smallest possible sum $a b$ is 7.

To learn more about the sum, refer:-

https://brainly.com/question/14643075

#SPJ4