What type of cooling solution is recommended for the Intel Pentium G6405, given its 58W power consumption?

The cooling device for the Intel Pentium G6405 includes a heatsink and fan.

Is the Intel Pentium G6405 compatible with motherboards that have an LGA 1200 socket?

Yes, the Intel Pentium G6405 is compatible with motherboards that have an LGA 1200 socket.

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Intel Pentium G6405 - Pentium Gold Comet Lake Dual-Core 4.1 GHz LGA 1200 58W Intel UHD Graphics 610 Desktop CPU Processor - BX80701G6405