Luke 10:4 (NASB)
Carry no money belt, no bag, no shoes; and greet no one on the way.
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Similarly, Elisha instructed his servant in 2 Kings 4:29
Elisha said to Gehazi, "Tuck your cloak into your belt, take my staff in your hand and run. Don't greet anyone you meet, and if anyone greets you, do not answer. Lay my staff on the boy's face."
Jesus sent his disciples out on a mission tour. He didn't want them to waste time and be sidetracked by friends or strangers. These are common tactics of Satan to delay or stall the mission. To avoid loitering and help them focus on the tasks at hand, Jesus commanded them not to perform these superficial activities.
There is also the story of the man of God from Judah in 1 Kings 13:
8 But the man of God answered the king, “Even if you were to give me half your possessions, I would not go with you, nor would I eat bread or drink water here. 9For I was commanded by the word of the Lord: ‘You must not eat bread or drink water or return by the way you came.’ ”
He obeyed, so far so good. But then later on, an old prophet
14 found him sitting under an oak tree and asked, “Are you the man of God who came from Judah?”
“I am,” he replied.
Big mistake! He socialized with him. Eventually,
24 as he went on his way, a lion met him on the road and killed him, and his body was left lying on the road, with both the donkey and the lion standing beside it.
To simplify the complications, it was best not even to greet anyone.