奧數(shù)題奇數(shù)個(gè)約數(shù)
寫(xiě)出從360到630的自然數(shù)中有奇數(shù)個(gè)約數(shù)的數(shù).
答案與解析:
如果某個(gè)自然數(shù)有奇數(shù)個(gè)約數(shù),那么這個(gè)數(shù)的'所有質(zhì)因子的個(gè)數(shù)均為偶數(shù)個(gè).這樣它們加1后均是奇數(shù),所得的乘積才能是奇數(shù).而所有質(zhì)因數(shù)的個(gè)數(shù)均是偶數(shù)個(gè)的數(shù)為完全平方數(shù).即完全平方數(shù)(除0外)有奇數(shù)個(gè)約數(shù),反過(guò)來(lái),有奇數(shù)個(gè)約數(shù)的數(shù)一定是完全平方數(shù)。即360到630的自然數(shù)中有奇數(shù)個(gè)約數(shù)的數(shù)為361,400,441,484,529,576,625.
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