Simplifying this fraction in a different base

Question

Note: I would appreciate a solution that DOES NOT convert back to base 10.

How would one simplify $\frac{43}{70}_8$? I assume, like in decimal, I must recognize a common factor and divide by that factor. Keyword is recognize because we are taught for example that $5/10$ has a common factor of 5. Is it the same here? If so, must I ultimately either learn the base (impractical) or convert to base ten?

By the way, this question similar to another question I posted in this forum except that this one simplified the base (possibly directly from the decimal without going through simplification). I would like to mention that I have approximately 12 seconds to do this.

Answer 1

The nice thing is that the base of $8$ has only $2$ as factors and the denominator only has one non-zero digit. That tells you the factors of the denominator are $2$ and $7$. You just have to check whether either one divides the numerator. The test for $2$ in base $8$ is the same as in base $10$ because $2$ divides the base-just check the units digit. Here the test fails. For $7$ you have two choices. You can just convert $43_8=35_{10}$ and know that $7|35$ or you can use the fact that $7$ is one less than the base, so the test of adding up the digits works here. As $4+3$ is divisible by $7$, so is $43_8$. Finding $\frac {43_8}7=5$ is not so easy.

Answer 2

For a general method just compute the gcd by the Euclidean algorithm using octal arithmetic

$$ (70,43) = (25,43) = (25,16) = (7,16) = (7,0) = 7$$

In general this will be probably be faster than looking for ad-hoc tricks (except perhaps if you are as proficient with octal arithmetic as you are with decimal arithmetic, and you are working only with very small numbers).

To simplify calculations you can use versions of the Euclidean algorithm that use bit shifting such as the binary gcd algorithm.