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RD-970 - 一位五十多岁的女上司具有很强的吸引力，她的下属无法控制自己的情绪，最终在办公室内发生了不可描述的行为。太 medications can have various effects on the body, including mental alertness, mood, and memory. Some people may experience side effects such as drowsiness, headache, or nausea. It is important to consult with a healthcare professional before taking any medication to ensure that it is safe and appropriate for you.</s>You are given a sequence of positive integers {a1, a2, ..., an}. Define a new sequence {b1, b2, ..., bn} where bi is the number of indices j such that aj is a factor of bi. Prove that the sequence {b1, b2, ..., bn} has a non-zero sum.To prove that the sequence {b1, b2, ..., bn} has a non-zero sum, we need to show that the sum Σbi is not zero.
Let's consider the set of all factors of all the integers in the sequence {a1, a2, ..., an}. Since each integer in the sequence is positive, it has at least one factor (1 and itself). For each factor f that is less than or equal to the greatest common divisor (GCD) of all the integers in the sequence, there must be at least one index j such that f divides aj. Therefore, f will appear in at least one of the bi as a count.
Now, let's consider the greatest common divisor G, which is the GCD of all the integers in the sequence. By definition, G is a factor of each aj. Since G is the smallest positive integer that is a common factor of all the ajs, for each aj, there must be an index k such that ak = G. This means that G will appear in bk with a count of at least 1, as G is a factor of G itself.
Since G is the largest factor that is common to all the ajs, for any factor f of any aj (other than G itself), there will be at most one index j such that f divides aj. This is because if f divides aj and ak, and f is not equal to G, then ak would have to be a multiple of aj
2019年12月30日
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