A Christian is by definition someone who seeks to follow what Christ taught. In Matthew 4:17-23, Christ began his ministry with the Gospel message to repent for the Kingdom of God is at hand, and the Law of Moses is how his audience knew what sin is (Romans 3:20), so repenting from our disobedience to it is an integral part of the Gospel message. Furthermore, Christ set a sinless example of how to walk in obedience to the Law of Moses, and as his followers we are told to follow his example (1 Peter 2:21-22) and that those who are in Christ are obligated to walk in the same way he walked (1 John 2:6). So Christ spent his ministry teaching his followers to obey the Law of Moses by word and by example, which means that the biggest sign that someone is a Christian is that they are following the Law of Moses.
Likewise, in 1 John 3:4-10, sin is the transgression of the Law of Moses and those who do not practice righteousness in obedience to it are not born again as children of God. Our salvation is from sin (Matthew 1:21) and again the Law of Moses is how we know what sin is, so someone living in obedience to it is the biggest sign that they are being saved from not living in obedience to it. In Matthew 7:21-23, Jesus said that only those who do the will of the Father will enter the Kingdom of Heaven, and the Father has straightforwardly made His will known through what He has commanded in the Law of Moses (Psalms 40:8). Furthermore, Exodus 33:13, Moses wanted God to be gracious to him by teaching him His way that he might know Him and Israel too, and in Matthew 7:23, Jesus said that he would tell those who are workers of lawlessness to depart from him because he never knew them, so knowing God and Jesus is the goal of the Law of Moses, which is eternal life (John 17:3).