Q: Consider two composite numbers A and B. Since A is composite we can write:

A = (p1^v1)(p2^v2)...(pn^vn)

Where p(i) is prime and v(i) is an integer. Suppose B is made up of the same primes, the difference is their exponent:

B = (p1^w1)(p2^w2)...(pn^wn) : w(i) is an integer

Show that A=B if and only if v(i) = w(i) for all i.

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A: (Necessary condition) Suppose v(i) = w(i). Clearly A=B.

But how can one deduce v(i) = w(i) given A=B as the starting point. How does one prove the sufficient condition?

My gratitude in advance to those bright sparks that prove this proposition.