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How many three-digit numbers \(ABC,\) in which \(A, B,\) and \(C\) are each digits, satisfy the equation \(2B = A + C?\)

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How many three-digit numbers \(ABC,\) in which \(A, B,\) and \(C\) are each digits, satisfy the equation \(2B = A + C?\)

by Gmat_mission » Sat Oct 31, 2020 6:38 am

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How many three-digit numbers \(ABC,\) in which \(A, B,\) and \(C\) are each digits, satisfy the equation \(2B = A + C?\)

A. 33
B. 36
C. 41
D. 45
E. 50

Answer: D

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Re: How many three-digit numbers \(ABC,\) in which \(A, B,\) and \(C\) are each digits, satisfy the equation \(2B = A +

by jhavyom » Sat Oct 31, 2020 11:57 pm
How many three-digit numbers ABC in which A, B, and C are each digits, satisfy the equation
2B = A + C

2B = A + C
This implies that the digits A, B, and C form an A. P.

For common difference (d) = 0,
ABC can be
111, 222, 333, 444, 555, 666, 777, 888, and 999
Total 9 digits with d = 0

For common difference (d) = 1,
ABC can be
123, 234, 345, 456, 567, 678, 789, 987, 876, 765, 654, 543, 432, 321, and 210
Total 15 digits with d = 1

Now, for d = 2
ABC can be
135, 246, 357, 468, 579, 975, 864, 753, 642, 531, and 420
Total 11 digits with d = 2

For d = 3,
ABC can be
147, 258, 369, 963, 852, 741, and 630
Total 7 digits with d = 3

For d = 4,
ABC can be
159, 951, 840
Total 3 digits with d = 4

Therefore, total 45 such numbers are there which satisfies 2B = A + C
Option D is the answer.
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