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It had a huge impact. Some would even say more than it has now. Out of the ten people that spoke that day, his speech was the most memorable and the one that really sparked the passion of the people.
Congressman John Lewis, who was the 6th speaker that day, describes MLK's words as "Streaks of lightning flying in the air." That speech and his notable role in the march on Washington contributed greatly to him getting a Nobel Peace Prize in 1964.