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The phrase divide and conquer is a common English idiom. Just like you said, it means to break a problem or task down into small pieces and then solve them one by one so that they feel more manageable than one big task. Alternatively, it can mean to divide work between members of a team so that no one has to do an overwhelming amount.
Divide and separate have the same meaning in that both create smaller parts from a greater whole, but the full context of how they are used and their needs to be understood to get the full meaning. Divide. Often means there is intent in making the disaggregation. The food was divided amoung the campers. On the other hand. Separate
divide 2 into 4, giving the answer 2. And in this definition: (of a number) be contained in a number without a remainder. "3 divides into 15" Then I guess that "divide 2 into 4 without a remainder" amounts to "2 divides into 4", and hence "the beats divide into twos" means "twos can be divided by the beats with no remainder". But does that ...
When half and two are used as adjectives, the correct phrasing is divide into. The math teacher divided the apple in two. The math teacher divided the apple into two parts. Mother divided the pile of toys in half. The team’s activities were divided into half days. Shall we divide the project into six or seven small tasks?
AFAIU, there's a phrase by Abraham Lincoln (and maybe Bible before that, I'm not sure, but it doesn't matter) 'a house divided against itself cannot stand'. I guess I somewhat understand the meani...
I think that you are correct with your assumptions, and where you are getting hung up is that divide is commonly used in the passive voice. For examples 1 and 3, it is natural to use the passive voice, as you have the sentences constructed. The US is divided into 50 states.
According to the Cambridge dictionary (image below), the phrasal verbs "break down" and "break up" share the same meaning, "to divide something into smaller parts". Moreover, according to this answer apparently there's a difference between these two phrasal verbs, but I'm not sure if that answer is correct.
'Sub-division' is usually used in relation to areas of land. Said areas have already been divided (E.g. separate 'lots' from a contiguous 'block'), so when somebody wishes to divide one of them further, they are said to be sub-dividing.
Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms? Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms? Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited" at the center.
You'll often see it used in anything conflict-related or where the splitting is forcefully uneven. However, it is also used in a more mathematical sense when you're splitting things into discrete sets, such as if you were to divide up pairs of socks. With regards to lines, divide is the word most often associated with them.