A researcher wants to insert a linear DNA fragment into a plasmid DNA. The linear DNA fragment has two MfeI restriction sites at its ends. The plasmid DNA has only one EcoRI restriction site. The plasmid was digested with EcoRI and the DNA fragment was digested with MfeI. The recognition sequences of the restriction enzymes on one strand are given.
EcoRI site: $5'\text{-G} \downarrow \text{AATTC-}3'$
MfeI site: $5'\text{-C} \downarrow \text{AATTG-}3'$
(arrow indicates enzyme cut sites)
Based on this information, the DNA fragments generated by EcoRI and MfeI

The restriction enzyme EcoRI recognizes the sequence $5'\text{-GAATTC-}3'$ and cuts between G and A, generating a $5'\text{-AATT-}3'$ sticky end.

The restriction enzyme MfeI recognizes the sequence $5'\text{-CAATTG-}3'$ and cuts between C and A, also generating a $5'\text{-AATT-}3'$ sticky end.

Since both enzymes produce identical $5'\text{-AATT-}3'$ overhangs, the DNA fragments have compatible sticky ends and can successfully ligate to each other.

When the fragments ligate, the junctions formed will be a combination of the half-sites from each enzyme:
Ligating the EcoRI left half-site ($5'\text{-G-}3'$) with the MfeI right half-site ($5'\text{-AATTG-}3'$) produces the sequence $5'\text{-GAATTG-}3'$.
Ligating the MfeI left half-site ($5'\text{-C-}3'$) with the EcoRI right half-site ($5'\text{-AATTC-}3'$) produces the sequence $5'\text{-CAATTC-}3'$.

The newly formed junction sequences are $5'\text{-GAATTG-}3'$ and $5'\text{-CAATTC-}3'$. Neither of these sequences matches the recognition site for EcoRI ($5'\text{-GAATTC-}3'$) or MfeI ($5'\text{-CAATTG-}3'$).

Therefore, the fragments can ligate to each other, but the resultant product can neither be digested by EcoRI nor MfeI.

Answer: can ligate to each other, but the resultant product can neither be digested by EcoRI nor MfeI.