Answer: What you've discovered are cabbage aphids; tiny greyish sucking insects. Like all aphids they multiply rapidly and seem impossible to control. You can wash the aphids off with a strong stream of water from the hose. Plan on daily treatment until you no longer see whole herds of them hiding under the leaves. Probably the most effective control would be ladybugs. If you import them, plan on covering your broccoli with a floating row cover and then releasing the ladybugs. That way they'll stay to do the job rather than flying away to your neighbors garden. If you don't want to purchase ladybugs, try growing a variety of flowers near your broccoli to attract them. They will visit for nectar and stay to consume the aphids.
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