Question

Determine the value of base b if (152)b = 0x6A. Please show all steps.

Answer 1

Answer:

The value of b is, 8

Step-by-step explanation:

Determine the value of b;

Given:

$(152)_b = 0x6A$             ....[1]

Since, 0x6A represents the hexadecimal form.

First convert this hexadecimal form into decimal form:

$(6A) = (6 \times 16^1)+(A \times 16^0) = (96)+(10 \times 1) = 96+10 = 106$

⇒$0x6A = 106$ (decimal form)

Now we have to convert this decimal form into octal

8  |  106

8   |  13   | 2

     |   1    | 5

          1

Then, the octal form we get, $(152)_8$

Substitute these in [1] we have;

$(152)_b=(152)_8$  

On comparing both sides we get;

b = 8

Therefore, the value of b is, 8

Answer 2

Assuming 0x6A is given in base 16, first convert $152_b$ and $6A_{16}$ to a common base, say base 10:

$152_b=1\cdot b^2+5\cdot b^1+2\cdot b^0=(b^2+5b+2)_{10}$

$6A_{16}=6\cdot16^1+10\cdot16^0=106_{10}$

Then

$b^2+5b+2=106$

$b^2+5b-104=(b-13)(b+8)=0$

$\implies b=13$