Why are my gums white around the teeth and achy after brushing my them?

The gum around your teeth should be pink in healthy condition but it is white as you are saying is sign of white lesion like Leukoplakia . If the gums are little white for your whole life then nothing to worry and just change the tooth paste and brush on teeth not on gums . It is better you See your Dentist to be sure it is nothing but normal.

What toothpaste or medicaments are you using on your toothbrush? Are you brushing really hard? When was your last dental cleaning? Answers to these questions would help in giving you a meaningful answer. See your dentist for an exam if it has been more than six months.

You may be brushing too hard or using too hard a t'brush. Or you may have an allergy to one of the ingredients in your t'paste. Or if you are not flossing your gums may be diseased. Or if you are not seeing your Dentist regularly, you may have unremoved tarter. See your Dentist for examination, professional cleaning, and review of your oral hygiene techniques.