Chaitanya's Random Pages

December 29, 2012

A curious fact involving factors

Filed under: mathematics — ckrao @ 10:07 pm

Here is one of my favourite elementary number theory results.

If a, b, c, d are positive integers such that ab = cd, then a + b + c + d is composite.

I like it because it is not immediately obvious that the relationship ab = cd should be related to the primality of the sum of these four numbers. As an example 12 = 2 \times 6 = 3 \times 4 and 2 + 6 + 3 + 4 = 15 = 5 \times 3.

To prove this, let g = gcd(a,c). Then we may write a = a_1 g and c = c_1 g where a_1 and c_1 have no common factors. In other words, g extracts the common factors from a and c. This leads to ga_1 b = g c_1 d or a_1 b = c_1 d. Since a_1 and c_1 have no common factors, this means all the non-unit factors of a_1 divide d (and all the non-unit factors of c_1 divide b). Then we may write d = a_1 h where h is a positive integer so that b = c_1 d / a_1 = c_1 h.

Then

\displaystyle a + b + c + d = a_1 g + c_1 h + c_1 g + a_1 h = (a_1 + c_1)(g + h)

which is composite.

Note that we can similarly prove that a^k + b^k + c^k + d^k is composite where k is any positive integer.

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