If a,b and c are positive integers where a is greater than b and c=(a)

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The original condition \(c=√a\) is equivalent to \(a = c^2\).
And it tells \(√a = c\) is an integer.

Condition 1)

\(\frac{a-b+1}{2} = c\)
\(⇔ a-b+1 = 2c\)
\(⇔ c^2-b+1 = 2c\), since \(a=c^2\)
\(⇔ c^2-2c+1 = b\)
\(⇔ (c-1)^2 = b\)
\(⇔ c-1 = √b\), since b and c are positive integers
\(⇔ c = √b+1\)
\(⇔ √a = √b+1\)
\(√a\) and \(√b\) are consecutive integers.

Condition 1) is sufficient.

Condition 2)
\(\frac{a+b-1}{8} = c\)
\(⇔ a+b-1 = 8c\)
\(⇔ c^2+b-1 = 8c\), since \(a=c^2\)
\(⇔ c^2-8c+b-1 = 0\)
\(⇔ c^2-8c+16 + b-1 = 16\)
\(⇔ c^2-8c+16 + b = 17\)
\(⇔ (c-4)^2 + b = 17\)

If \(c = 4\) and \(b = 17\), then we have \(a = c^2 = 16, b = 17\) and the answer is 'yes'.
If \(c = 8\) and \(b = 1\), then we have \(a = c^2 = 64, b = 1\) and the answer is 'no'.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.