identify the base

Question

Identify the base in which the following subtraction problem is correct: 

 

52___ - 15___ =35 ___

Answer 1

• Clearly it's not Base 10 since 52-15 = 37 in our conventional base 10 system
• But, it must be something close to base 10 since 37 almost equals 35
• I'm not sure there is a way to solve this other than to experiment with different bases until the equation is true.
• Try base 9
• 52₉ - 15₉ =   36₉.  So it is not base 9
• Check your work by converting the numbers to base 10.  52₉ = 47₁₀ and 15₉ = 14₁₀.  So 47-14 = 33₁₀.  Does 36₉ = 33₁₀?  Yes.  Why?  Because (3*9¹) + (6*9⁰) = 27₁₀ + 6₁₀ = 33₁₀
• Try base 8
• 52₈ - 15₈ = 35₈.   The equality is true in base 8.
• Check your work by converting the numbers to base 10.  52₈ = 42₁₀ and 15₈ = 13₁₀.  So 42-13 = 29₁₀.  Does 35₈ = 29₁₀ ?  Yes.  Why?  Because (3*8¹) + (5*8⁰) = 24₁₀ + 5₁₀ = 29₁₀