Hi i did review your concern. the purple bruise you are talking about could be a birth mark known as mongolian spot. please ask your mother if its from childhood or recently noticed. if noticed recently you should consult a dermatologist to find out more about it. Morever, the back pain could be because of multitude of causes, primary could be arthritis itself or a disk herniation or lumbar spinal stenosis. does the pain occur more walking downhill or is it relieved by bendin forward. check if she feels pain on walking on toes.if yes, she could be having lumbar spinal stenosiss common at this age. You can then probably go for an x ray and an MRI to know finer details of the back bones and muscles and spinal cord involvement. I hope this helps wish you a healthy life