The value of the two-digit number aβ bβ in base seven equals the value of the two-digit number bβ aβ in base nine. What is a+b?
Answer Choices:
A. 7
B. 9
C. 10
D. 11
E. 14
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The base-7 number ab7β equals the base-9 number ba9β:
7a+b=9b+a.
Thus
7a+b=9b+aβΉ6a=8bβΉ3a=4b.
So a=4k, b=3k for some integer k. Since a,b are digits in base 7, we require 1β€a,bβ€6. The only possibility is k=1, giving (a,b)=(4,3).
Hence
a+b=4+3=(A) 7β.
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