It's a good question and my answer is that the old testament law does not save, it shows that we are sinners. Romans 2:
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Romans 2
New International Version
God’s Righteous Judgment
2 You, therefore, have no excuse, you who pass judgment on someone else, for at whatever point you judge another, you are condemning yourself, because you who pass judgment do the same things. 2 Now we know that God’s judgment against those who do such things is based on truth. 3 So when you, a mere human being, pass judgment on them and yet do the same things, do you think you will escape God’s judgment? 4 Or do you show contempt for the riches of his kindness, forbearance and patience, not realizing that God’s kindness is intended to lead you to repentance?
5 But because of your stubbornness and your unrepentant heart, you are storing up wrath against yourself for the day of God’s wrath, when his righteous judgment will be revealed. 6 God “will repay each person according to what they have done.”[a] 7 To those who by persistence in doing good seek glory, honor and immortality, he will give eternal life. 8 But for those who are self-seeking and who reject the truth and follow evil, there will be wrath and anger. 9 There will be trouble and distress for every human being who does evil: first for the Jew, then for the Gentile; 10 but glory, honor and peace for everyone who does good: first for the Jew, then for the Gentile. 11 For God does not show favoritism.
12 All who sin apart from the law will also perish apart from the law, and all who sin under the law will be judged by the law. 13 For it is not those who hear the law who are righteous in God’s sight, but it is those who obey the law who will be declared righteous. 14 (Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law. 15 They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them.) 16 This will take place on the day when God judges people’s secrets through Jesus Christ, as my gospel declares.
As the others have said Jesus is the fulfillment of the law and its ultimate sacrifice for sins.
As Hebrews 1says:- 1 In the past God spoke to our ancestors through the prophets at many times and in various ways, 2 but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe. 3 The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word. After he had provided purification for sins, he sat down at the right hand of the Majesty in heaven. 4 So he became as much superior to the angels as the name he has inherited is superior to theirs.
Hebrews shows that Jesus dealt with our sins, a other quote shows even more graphically why we are Christ ians and not Jews.
1 Cor 15:14 And if Christ has not been raised, our preaching is useless and so is your faith. 15 More than that, we are then found to be false witnesses about God, for we have testified about God that he raised Christ from the dead. But he did not raise him if in fact the dead are not raised. 16 For if the dead are not raised, then Christ has not been raised either. 17 And if Christ has not been raised, your faith is futile; you are still in your sins. 18 Then those also who have fallen asleep in Christ are lost. 19 If only for this life we have hope in Christ, we are of all people most to be pitied.
In this Paul shows that if Jesus didn't rise from the dead we have no hope of heaven, only condemnation as sinners.
Hope this long set of quotes helps.
. In Christ any human being can be forgiven washed and made clean 
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