12 In this way, it is the taking of our sins upon Himself by the Lord Christ that is signified in the genealogy of Matthew, while in the genealogy of Luke it is the abolition of our sins by the Lord Christ that is expressed. In accordance with these ideas, the one details the names in the descending scale, and the other in the ascending. For when the apostle says, “God sent His Son in the likeness of the flesh of sin,” he refers to the taking of our sins upon Himself by Christ.
But when he adds, “for sin, to condemn sin in the flesh,” he expresses the expiation of sins. Consequently Matthew traces the succession downwards from David through Solomon, in connection with whose mother it was that he sinned; while Luke carries the genealogy upwards to the same David through Nathan, by which prophet God took away his sin. The number, also, which Luke follows does most certainly best indicate the taking away of sins. For inasmuch as in Christ, who Himself had no sin, there is assuredly no iniquity allied to the iniquities of men which He bore in His flesh, the number adopted by Matthew makes forty when Christ is excepted.
On the contrary, inasmuch as, by clearing us of all sin and purging us, He places us in a right relation to His own and His Father's righteousness (so that the apostle's word is made good: “But he that is joined to the Lord is one spirit”), in the number used by Luke we find included both Christ Himself, with whom the enumeration begins, and God, with whom it closes; and the sum becomes thus seventy-seven, which denotes the thorough remission and abolition of all sins. This perfect removal of sins the Lord Himself also clearly represented under the mystery of this number, when He said that the person sinning ought to be forgiven not only seven times, but even unto seventy times seven.
Source: The Harmony of the Gospels (New Advent)