A 3-digit number in base 7 has its digits reversed when expressed in base 9.

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Subject: Converting between bases
Name: Ashley
Who are you: Student

A 3-digit number in base 7 has its digits reversed when expressed in base 9. Find this number.

Could you please help?

Hi Ashley,

I can get you started.

Suppose that

abc₇ = cba₉

First notice that since abc₇ is a legitimate base seven number, a, b and c are all non-negative integers, less than 7. Now I want to convert both numbers to base 10, thus

7² a + 7 b + c = 9² c + 9 b + a

(7² - 1) a + (7 - 9) b + (1 - 9²) c = 0
48a - 2b - 80c = 0

Dividing through by 2 gives

24a - b - 40c = 0

8 divides both 24 and 40 so 8 divides b. But b is a non-negative integer less than 7 so b = 0.

Can you finish it now?
Penny

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