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The component primarily responsible for carrying out calculations in a computer is the CPU, also known as the Central Processing Unit. The CPU is often referred to as the "brain" of the computer because it executes instructions from programs by performing arithmetic, logical, control, and input/output operations specified by those instructions.

The CPU operates with a set of registers and uses its arithmetic logic unit (ALU) to perform complex calculations. It processes data and manages tasks by quickly retrieving and interpreting information stored in the RAM and other components. This makes it the central hub through which all the computational power of the machine is realized.

In contrast, the hard drive serves as storage for data and programs and does not directly participate in calculations. RAM, while crucial for temporarily holding data that the CPU needs to access quickly, also does not perform calculations itself. The motherboard connects all components, including the CPU, but it does not perform calculations. Therefore, the CPU stands out as the definitive component that carries out the actual computational tasks in a computer system.