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| all mainland Chinese, as well as Taiwanese, hongkongese and overseas Chinese speak Mandarin to each other | |
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No they don't. Older people in HK don't speak Mandarin, neither do many overseas Chinese who left HK prior to around 1987, or who left mainland China a generation or so ago. There are quite a few Chinese in Singapore and Malaysia who can speak Hokkien but not Mandarin, for example.
But yes, when someone says "I'm from China and will teach you Chinese" it's going to be Mandarin.