ābc and cbā are respectively ,the base 9 and the base seven numerals for the same positive integer. find the sum of digits of this s integer when expressed in base ten.
Answers
Given : ābc and cbā are respectively ,the base 9 and the base seven numerals for the same positive integer.
To find: The sum of digits of this integer when expressed in base ten.
Solution:
- Now we haeve given that ABC is base 9 and CBA is base 7.
- Consider that X(10) = (ABC)(9) = (CBA)(7)
- Lets take X(10) = (ABC)(9), we get:
= 9^2(A) + 9^1(B) + 9^0(C)
= 81A + 9B + C .........................I
- Lets take X(1)0 = (CBA)(7), we get:
= 7^2(C) + 7^1(B) + 7^0(A)
= 49C + 7B + A .........................II
- Now equating I and II, we get:
81A + 9B + C = 49C + 7B + A
2B = 48C - 80A
B = 24C - 40A
B = 8(3C - 5A)
- So here A, B & C are in base 7 so A , B , C is less than 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)
- Now we have:
X(10) = 81A + 9B + C
= 81(3) + 0 + 5
= 248
- or , we have:
X(10) = 49C + 7B + A
= 49(5) + 0 + 3
= 248
(305)(9) = (503)(7) = (248)(10)
- So, sum of digits of this integer when expressed in base ten is:
(248)(10) = 2 + 4 + 8 = 16
Answer:
The sum of digits of this integer when expressed in base ten is 16.
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