Too many bases

Number Theory Level 2

${ \left( 01 \right) }_{ 2 }+{ \left( 34 \right) }_{ 5 }+{ \left( 67 \right) }_{ 8 }\quad =\quad { \left( \overline { AB } \right) }_{ 9 }$

What are the digits $A$ and $B$ that satisfy the above relation?

Details and Assumptions:

$\left( 34 \right)_5$ represents a number, say $34$ , in base $5$

74 83 47 65

1 solution

Ralph James
Apr 2, 2016

We know that $0 \le A,B \lt 10$ and that $A,B \in \mathbb{Z^+}$ .

$01_{2}=0 \cdot 2 + 1=1_{10}$
$34_{5}=3 \cdot 5 + 4 = 15 + 4 = 19_{10}$
$67_{8}=6 \cdot 8 + 7 = 48 + 7 = 55_{10}$ $55 + 19 + 1 = 75_{10} =\overline{ AB }_{9} = 9A + B \implies 9 \mid 75 - B$ $A = \frac{75 - B}{9}$ $75 - 3 = 72, \frac{72}{9} = 8 \implies AB = \boxed{83}$