Determine all composite integers n>1 that satisfy the following property:
if d1,d2,⋯,dk are all the positive divisors of n with 1=d1<d2<⋯<dk=n, then di divides di+1+di+2 for every 1≤i≤k-2.
译文:设1=d1<d2<⋯<dk=n是合数n的全部正因数,若对任意1≤i≤k-2,有di |di+1+di+2,求n.
Determine all composite integers n>1 that satisfy the following property:
if d1,d2,⋯,dk are all the positive divisors of n with 1=d1<d2<⋯<dk=n, then di divides di+1+di+2 for every 1≤i≤k-2.
译文:设1=d1<d2<⋯<dk=n是合数n的全部正因数,若对任意1≤i≤k-2,有di |di+1+di+2,求n.
∵n是合数,∴k≥3.先证:d2 |di,i=2,3,⋯,k.∵di |dk (i=1,2,3,⋯,k),且dk-2 |dk-1+dk,∴dk-2 |dk-1,又∵d2∙dk-3=d3∙dk-2=n,∴n/d3 |n/d2 ,∴d2 |d3,又∵d2 |d3+d4,∴d2 |d4.同理推得:d2 |d5,⋯,d2 | dk,即d2 |di,i=2,3...
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