Jesus said this on divorce: “It was said, ‘Whoever sends his wife away, let him give her a certificate of divorce’; 32 but I say to you that everyone who [a]divorces his wife, except for the reason of unchastity, makes her commit adultery; and whoever marries a [b]divorced woman commits adultery." (Mt 5:31-32 NASB)
Notice that the text has markers [a] and [b]. Why? In this case, the text was taken from Biblegateway's NASB translation. Those markers indicate that the translators have made a change to the text, and the word used is not actually "divorce." That should make you stop and think. So what is the proper translation? Let's look at the literal of v32:
"But I say unto you, that every one that putteth away his wife, saving for the cause of fornication, maketh her an adulteress: and whosoever shall marry her when she is put away committeth adultery."
Jesus was speaking to Jews, so we should go the the Jewish Law on divorce to understand what He was expressing. Specifically, Deut 24:1-4 informs us about the Law on divorce. v1-2 say,
“When a man takes a wife and marries her, and it happens that she finds no favor in his eyes because he has found some indecency in her, and he writes her a certificate of divorce and puts it in her hand and sends her out from his house, and she leaves his house and goes and becomes another man’s wife."
Divorce was accomplished through a legal paper of divorcement, known as a "get." Without that paper, a woman was still married, and so lying with another man was adultery. Sending a wife away without a "get" was known as "putting away," and was a terrible thing to do to a woman, as it stripped her of her ability to remarry. The New and Old Covenants both taught that those who are not married (including someone legally divorced) are allowed to remarry.