How to put a fraction in simplest form, such as 140 255 ?

Question

How to put a fraction in simplest form, such as             140      255      ?

Lisbonaid · Accepted Answer

Like the man who invented 5-Up and 6-Up, you stopped one number short of success. Of 140 and 255, only the former is divisible by 2 and by 4, and only the latter is divisible by 3, but both are divisible by 5. This should immediately lead to the answer:

\frac{28}{51}

How can you find a common factor? If none immediately suggest themselves, you can try getting the prime factorization of the numerator and the denominator, and this works reasonably fine for small numbers like 140 and 255.
For anything much bigger than that, you might need the Euclidean GCD algorithm. Take for example

\frac{279244}{508623} .

Your computer can immediately see the common factor, but without a computer (e.g., on a test for which you're only allowed a basic calculator), you don't want to spend too much time on trial division. So:

508623 = 1 \times 279244 + 229379

279244 = 1 \times 229379 + 49865

229379 = 4 \times 49865 + 29919

49865 = 1 \times 29919 + 19946

29919 = 1 \times 19946 + 9973

19946 = 2 \times 9973 + 0

On second thought, I hope you don't get a test problem where the GDC is a 4-digit prime.

abbracciopj · Accepted Answer

To put a fraction       a    b   in simplest form, the most straightforward way usually comes from knowing   gcd  (  a  ,  b  ). In the case of       140    255   , we have   gcd  (  140  ,  255  )  =  5. Divide the numerator by 5 and divide the denominator by 5 to obtain      28    51    .To verify the answer, check that   gcd  (  a  ,  b  )  =  1

How to put a fraction in simplest form, such as

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