prove that if a q1e1 q2e2 qrer and b s1f1 s2f2 sufu ar

prove that, if a = q1^e1 q2^e2 ... qr^er and b = s1^f1 s2^f2 ... . . su^fu are the factorizations of a and b into primes, then there exists primes t1<t2<t3<tv and non negative integers gi and hi such that a = t1^g1 t2^g2 ....tv^gv and b = t1^h1 t2^h2 ....tv^hv

Solution

Let t₁<t₂<t₃<...<t_v, to be ALL primes less than (a×b).

For each t_k:
If t_k divides a: Find the largest power of \"t_k\", call it \"g_k\", which divides \"a\". So t_k^g_k divides \"a\", but t_k^(g_k+1) does not.

Why? Becuase t_k^(g_k+1) = tk^gk. t_k and gk is the largest power that divides a.
If t_k does not divide a, then take g_k = 0, which is non-negative so that t_k^g_k=1 still divides a: gk is still the largest power of tk that divides a.

Each of t_k^g_k divides a since we find all possible factors.

In fact the product t1g1. t2g2.t3g3... tkgk= a as all the powers g1, g2, g3,... gk are co-prime

Repeat the steps to find non-negative h_k and form b .