AMC 10B/12B 2025 (Problem 4/4)The value of the two-digit number a‾ b‾\underline{a}\ \underline{b}a b in base seven equals the value of the two-digit number b‾ a‾\underline{b}\ \underline{a}b a in base nine. What is a+ba+ba+b?(A) 7\text{(A)}\;7(A)7(B) 9\text{(B)}\;9(B)9(C) 10\text{(C)}\;10(C)10(D) 11\text{(D)}\;11(D)11(E) 14\text{(E)}\;14(E)14Related TopicsCoreToolkit 104 — Number Base Representations and ConversionsHints (4)Hint 1a‾ b‾7=b+7a⇒1≤a≤6,0≤b≤6\underline{a}\ \underline{b}_{7}=b+7a\qquad\Rightarrow\qquad 1\le a\le6,\quad 0\le b\le6a b7=b+7a⇒1≤a≤6,0≤b≤6Hint 2b‾ a‾9=a+9b⇒1≤b≤8,0≤a≤8\underline{b}\ \underline{a}_{9}=a+9b\qquad\Rightarrow\qquad 1\le b\le8,\quad 0\le a\le8b a9=a+9b⇒1≤b≤8,0≤a≤8Hint 3By Hints 1 and 2b+7a=a+9b1≤a,b≤6b+7a=a+9b\qquad 1\le a,b\le6b+7a=a+9b1≤a,b≤6⇒6a=8b\Rightarrow 6a=8b⇒6a=8b⇒3a=4b\Rightarrow 3a=4b⇒3a=4bHint 43a=4b1≤a,b≤63a=4b\qquad 1\le a,b\le63a=4b1≤a,b≤6⇒a is a multiple of 4⇒a=4\Rightarrow a\text{ is a multiple of }4\Rightarrow a=4⇒a is a multiple of 4⇒a=4⇒12=4b⇒b=3⇒a+b=4+3=7\Rightarrow 12=4b\Rightarrow b=3\Rightarrow a+b=4+3=7⇒12=4b⇒b=3⇒a+b=4+3=7Final Answer(A) 777