4. Let a, p and q be positive integers with p≤q. Prove that if one of the numbers ap and aq is divisible by p , then the other number must also be divisible by p .
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03.02.2013 18:06
p cannot have more than p of any prime factor, so if a contains all of the prime factors of p then certainly ap and aq do, otherwise not.
03.05.2013 21:28
Hi ; I Think The Other Number Must Divisible By q Best Regard
24.01.2014 22:29
Let a=∏ri=1qiαi and p=∏sj=1pjβj Now p divides ap or aq means (pj)sj=1 is a subset of (qi)ri=1.Also the exponent βi of any pi cannot exceed p since for some i if it exceeds p then p=∏sj=1pjβj≥piβi≥pip≥2p>p a contradiction. It readily follows that p|ap⇔p|aq. Such a bad problem.
25.01.2014 01:05
We will show that if p∣ap or p∣aq, then it forces p∣ap, and since ap∣aq it will also be p∣aq. The case p=1 being trivial, assume p≥2. Consider any prime t dividing p, so it must also divide a. Let u≥1 be the exponent of t in p and v≥1 be the exponent of t in a. But for any values of u and v we have u<tu≤p≤vp, so tu∣(tv)p. This means p∣ap.