Question

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

Answer 1

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`

Answer 2

The value of b is, 8

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 * 16^1)+(A * 16^0) = (96)+(10 * 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