The largest three digits is 999, divided by 53, the quotient is 18, and the remainder is 45, that is, the largest quotient of three digits divided by 53 is 18. However, the maximum remainder of a number divided by 53 is 52, which is 7 larger than 45, so only 999-53+7 = 953,953 ÷ 53 =17 ... 52, that is, a+b= 17+52=69 is the maximum value.