Numbers in a Different Base!

Algebra Level 5

I take a three-digit base-ten integer (in which the first digit is non-zero) and consider it as a number in a different base. If I convert this new number into base ten, I find that it is exactly twice the original number. In what base does this happen?


The answer is 15.

Satyajit Mohanty by Satyajit Mohanty , 24, India

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1 solution

Bob Kadylo
May 25, 2017

Let x = x= the base we are looking for and a b c ‾ = \overline{abc}= the original base 10 number.

a, b, c are base 10 digits with 1 ≤ a ≤ 9 , 0 ≤ b ≤ 9 , 0 ≤ c ≤ 9 1\leq a \leq 9 , 0\leq b \leq 9 , 0\leq c \leq 9

To be solved: a x 2 + b x 1 + c x 0 = 2 ( 100 a + 10 b + c ) ax^2+bx^1+cx^0=2(100a+10b+c) in positive Integers.

I found 3 solutions for { a , b , c , x } → { 1 , 4 , 5 , 15 } , { 1 , 5 , 0 , 15 } , { 2 , 9 , 5 , 15 } \lbrace{a,b,c,x\rbrace}\rightarrow\lbrace{1,4,5,15\rbrace},\lbrace{1,5,0,15\rbrace},\lbrace{2,9,5,15\rbrace}

Answer: 15 \boxed{15}

Note: solving the diophantine equation is left as an exercise since my limited familiarity with LaTeX precludes my showing it here.

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