設(shè)正整數(shù)分解質(zhì)因數(shù)P1^a1*P2^a2*...*Pn^an,
則它的約數(shù)的個(gè)數(shù)為(a1+1)(a2+1)(a3+3)...(an+1)
因?yàn)轭}中要求的數(shù)能被30整除,所以必然含有質(zhì)因數(shù)2,3,5,
設(shè)此數(shù)為2^a1*3^a2*5^a3.
則它的約數(shù)個(gè)數(shù)為(a1+1)(a2+1)(a3+1),
又因?yàn)?0=2*3*5,所以此數(shù)沒(méi)有除2,3,5之外的質(zhì)因數(shù),所以a1+1,a2+1,a3+1只能是2,3,5或者3,2,5或5,2,3或5,3,2或2,5,3或3,5,2,共有6個(gè).
則a1,a2,a3只能取1,2,4或2,1,4或4,1,2或4,2,1或1,4,2或2,4,1.
即該6個(gè)自然數(shù)分別為
2^1*3^2*5^4=11250
2^2*3^1*5^4=7500
2^4*3^1*5^2=1200
2^4*3^2*5^1=720
2^1*3^4*5^2=4050
2^2*3^4*5^1=1620
所以綜上所述,最大自然數(shù)是11250