Find the Base

Number Theory Level 2

123 10 = 323 b \huge{123}_{10}={323}_{b}

Find the positive integer base b b .

Clarification: In what base does the base-10 numeral 123 equal 323?

7 8 2 9 6 5 3 4

View wiki
Arulx Z by Arulx Z , 20, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Arulx Z
Feb 4, 2016

32 3 b 323_b can be written as 3 b 2 + 2 b + 3 3b^2+2b+3 . Since 32 3 b = 123 323_b = 123 , this can be formulated into a quadratic equation -

3 b 2 + 2 b + 3 = 123 3 b 2 + 2 b − 120 = 0 ( 3 b + 20 ) ( b − 6 ) = 0 3{ b }^{ 2 }+2b+3=123\\ 3{ b }^{ 2 }+2b-120=0\\ \left( 3b+20 \right) \left( b-6 \right) =0

So b = 6 b = 6 or b = − 20 3 b = -\frac{20}{3} . Since the base is a positive integer, the answer is b b .

16 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Simple standard approach.

However, it is not true that a base cannot be negative, or fractional, or even irrational.

@Calvin Lin That's true and I indeed noticed that but I have seen that traditionally people neglect those roots. Anyways, for the sake of clarity, I have added a warning.

Arulx Z - 5 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...