Muddle becomes major event. How do I settle this?

My mother had a phone call out of the blue from A. Mother asked A who gave her the phone number. A said it was B, her son, my brother. Mother was cross as she does not have B's own phone number (he writes nice letters quite frequently and calls her about twice a year but for some reason won't give her his number and has asked me not to). She felt he had no right in these circumstances to give out her number without permission. She became increasingly concerned about this and asked me (her daughter and LPA) to email my brother and say she was cross and would he ask permission first. I did this politely and suggested (as there have been issues between B and our mother ) that he might want to say sorry for the mistake to prevent any further trouble between them. B went ballistic. Emailed me loads of abuse, called me a liar, wicked, used weird over-the-top language in reply to a polite and friendly email. He said I had made this up, was trying to cause trouble and that mother loved him and would never be cross with him about this or anything else - and anyway, he did not give A the number it was someone else: C. He contacted A and got her to swear she didn't tell mother he gave her the number, that it was C, and to say that mother must have misheard her on the phone. Because of a huge unpleasant muddle now on the family grapevine with C getting the blame, my mother has has now denied asking me to pass on the initial message and says she doesn't mind who has her phone number! Fine, I'm prepared to weather being the family baddie for the time being but my mother, it seems, isn't going to rest until I apologise to my brother and to A and C for making trouble. She expects me to smooth it all out and she is waiting to hear from them that I've done this. I'm unsure how to soothe mother who certainly didn't expect all this trouble any more than I did (I should have let her write to him herself, I realise this now). She is obsessing about it and keeps phoning people up to ask what is happening now - what's the situation etc etc. NB: the abuse I had emailed to me was so vitriolic and was such an out of the blue shock that I had a bad angina attack which actually prevented me from visiting mother as planned this week. Neither A or B ever visit mother but they do write to her which gives her pleasure. I would like to settle this without sacrificing my own integrity which I am not prepared to do. Sensible and kind solutions welcomed!