bar(ABC) and bar(CBA) are, respectively, the base nine and base seven numerals for the same positive integer. Find the sum of digits of this integer when expressed in base ten.
Answers
Given : (ABC) and (CBA) are, respectively, the base nine and base seven numerals for the same positive integer
To find : sum of digits of this integer when expressed in base ten
Solution:
ABC is base 9
CBA is base 7
Let say N₁₀ = (ABC)₉ = (CBA)₇
N₁₀ = (ABC)₉
= 9²*A + 9¹*B + 9⁰*C
= 81A + 9B + C
N₁₀ = (CBA)₇
= 7²*C + 7¹*B + 7⁰*A
= 49C + 7B + A
81A + 9B + C = 49C + 7B + A
=> 2B = 48C - 80A
=> B = 24C - 40A
=> B = 8(3C - 5A)
A, B & C are in base 7
hence A , B , C < 7
B = 8(3C - 5A) < 7
=> B = 0
=> 3C - 5A = 0
=> C = 5 , A = 3
A = 3 , B = 0 C = 5
(305)₉ = (503)₇
N₁₀ = 81A + 9B + C = 81*3 + 0 + 5 = 248
or N₁₀ = 49C + 7B + A = 49*5 + 0 + 3 = 248
(305)₉ = (503)₇ = (248)₁₀
sum of digits of this integer when expressed in base ten.
(248)₁₀
= 2 + 4 + 8
= 16
16 is the Sum of digits of this integer when expressed in base ten.
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