How do electrons divide? Electrons show a banded distribution at the end of a double-slit experiment set-up. This banded pattern shows that wave interference prevents many electrons from reaching areas where probability is lowest.
My question is about how an electron can pass through both slits without dividing because electrons are detected in whole at the end of the experiment.
How does a single object pass through two separate areas (the two slits) at once?
 A: A wave function is not a particle until it is detected.
The wave function relates to the relative likelihood of detecting the particle at each point & time.
Whereas a wave function is spread out in both space and time, a particle can have a definite position or momentum at a given moment.
So, the electron does not divide.  The wave function divides.  The wavefunction behaves as a wave and so can pass through both slits- as long as the detection method leaves the trajectory of the electron indeterminate.
A: It takes an aura of mystery to get enough attention. The interpretation of the double-slit experiment is such a thing. "It is beyond our imagination, it is quantum mechanical" is the wonderful sentence that makes us think and always ask questions.
The answer of PhysicsDave is the one that is different from the other answers:

The electron does not split into 2 parts ... it chooses one slit or the other. The wave properties are a result of the EM field which influences all charge particle trajectories. It is the wave property that results in the banding (not the interaction of 2 electrons in some kind of superposition).

This is the key. You have the electron with its electric and its magnetic field and you have the edge (of the slit(s)) with its field.To understand it, remember that under certain technical circumstances we are able to couple EM radiation into the skin surface of a material and obtain surface plasmons. What you get are surface plasmon polariton waves, and these waves can exit the material and radiate again.
The wonderful mystery of the slit experiments includes the refusal to consider alternatives. The alternative is simple. The particle interacts with the edge region in a quantized way and the deflections lead to the intensity distribution on the screen.
More information can be found in the answer to the question How does the Huygens principle explain interference?.
A: Because, although it is a single object, it is a wave and thus not a local object which can be described by a point moving in space, but rather a non local thing: a wave which moves through space and therefore passes both slits simultaneously. When one tries to describe an electron and perform an experiment to detect it as a such one loses the interference pattern.
A: Because an electron is not just a particle, it is also a wave. This is is known as wave-particle duality. So if you think of an electron as a wave, you can think of it kind of like a wave of water going through two slits, it will create an interference pattern on the screen. Here is a video hosted by Brian Greene that gives it a good visual.
A: To say a substantially similar thing to the existing answers in very different words, quantum mechanics is now known to be a deeply nonlocal theory: you cannot carve the world up into subsystems moving around and containing hidden classical information obeying a classical probability model, without allowing that information to affect other information in other subsystems instantaneously at a distance.
It is a "relativistic" nonlocality, which means that the nonlocality can only be observed in correlations between two different experiments but those experiments' results need to be brought back together (by classical processes) into one place to detect that nonlocality. This forbids using QM to observably transfer information any faster than the speed of light. Sometimes this is explicit, so for example in "quantum teleportation" one party must send a couple bits of classical information to the other party for them to "descramble" the teleported state and then use it normally.
When Feynman says that the double slit experiment contains the heart of quantum mechanics, I think this is really what he is driving at. He is saying that this aspect of nonlocality is present even here. So for example the pilot-wave interpretations of quantum mechanics have the particle go through one or the other of the slits, but the fact that it could have gone through the other causes its pilot wave to pass through the other slit, and the interference on the pilot wave causes the electron to not hit the various "dead zones" on the detector screen. The pilot wave is itself a manifestation of this nonlocality, the electron goes through one slit but it can nonlocally "see" that the other slit was available to go through.
My favorite demonstration of this nonlocality is a game for a team of 3 people to try to beat us scientists. We, the people setting up the game, split the team into 3 rooms and then either set up a "control round" where the three are all given the same goal, call it Goal A, or "betrayal rounds" where one person is given this goal A but the other two are told to accomplish the opposite, call it Goal B, so that the one person is unwittingly "betraying" the other two because they have the wrong information about the goal. The 3-person team has to somehow detect and correct for this "traitor" if they want to win the round and accomplish the "true" goal B; but we can relativistically separate the rooms so that they cannot classically communicate with each other. Classical teams can satisfy at most 3 of 4 possible equations, so if we set up those 4 possibilities to be equally likely, they can only win a maximum of 75% of the time. Teams who share a quantum state can cooperate to win the game 100% of the time in theory -- today they would be limited sharply by our ability to keep these delicate "entangled states" truly "coherent" in ways that would allow them to actually win 100% of the time.
But the point is, we know the classical picture quite exhaustively; the classical picture of probability allows you to transform the above problem into one where we say to all three team members separately, "okay if we gave you goal A what would you answer? and if we gave you goal B what would you answer?" and then afterwards look at all four setups AAA, ABB, BAB, BBA and choose each one with a 25% probability, and we would have a 25% chance of hitting the one that they did not choose. The experimental-setup choice of "what situation are we setting up?" can "commute" with the "what is your strategy?" choice. In quantum mechanics, this commutation cannot happen in this way.
A: In the ensemble interpretation the wave function describes an ensemble of particles rather than a single one. Some electrons go  through one slit and others through the other. It does not solve the mystery of the wave function and the probabilistic nature of QM, but it avoids this and similar paradoxes.
A: An electron is a particle with wave properties.  The electron does not split into 2 parts ... it chooses one slit or the other.  The wave properties are a result of the EM field which influences all charge particle trajectories. It is the wave property that results in the banding (not the interaction of 2 electrons in some kind of superposition).
IMO the electron behaves similarly to Feynman interpretation of a photon in the DSE. Thus the electron is most likely to travel a path length n times its wavelength....and this in turn allows/favours certain paths while disallowing others.  The emitting atom/electron field combined with the receiving atom/electrons field are optimal for optimal path length.
