What is the lowest term of 96 over 1440?

Answer 1

The question is: "How do you bring 96/1440 to lowest terms?"

First, you find the greatest common divisor of 96 and 1440.

If we write (n,m) for the greatest common divisor, then we have the following rules:

1. (n,m) = (m,n), symmetry

2. (n,m) = (n-m,m), subtract one coordinate from the other

3. (n,m) = (n+m,m), add one coordinate to the other

4. (n,0) = n

Proof of 1. Trivial.

Proof of 2. Let k be a common divisor of n and m, such that n=k.r and m=k.s, then n-m = k.(r-s) is divisible by k and m is divisible by k, so all common divisors of n and m are common divisors of n-m and m.

Proof of 3. same idea as 2. Left as an exercise.

Proof of 4. Left as an exercise.

Hence, (96,1440) = (96,0) = 96.

Second, you find the quotient using the greatest common divisors and calculate the smallest terms, thus:

96/1440 = (96.1)/(96.15) = 1/15 in smallest terms.

<Feb 18, 2014 - B.C.>