Question

A, B, C, D, E and F are six positive integers such that
\(B + C + D + E = 4A\)
\(C + F = 3A\)
\(C + D + E = 2F\)
\(F = 2D\)
\(E + F = 2C + 1\)
If \(A\) is a prime number between 12 and 20, then which of the following must be true?

• A. D is the lowest integer and D = 14
• B. C is the greatest integer and C = 23
• C. B is the lowest integer and B = 12
• D. F is the greatest integer and F = 24

Hint:

Solve the system fully first, then sort all six values from smallest to largest before checking each statement.

Answer 1

The Correct Option is C

Step 1: Recall the full solved set of values.
Solving the system, as in the earlier two questions of this set, gives the unique solution \(A = 17\), \(B = 12\), \(C = 23\), \(D = 14\), \(E = 19\), \(F = 28\), the only set of positive integers that satisfies all five equations with \(A\) prime between 12 and 20.

Step 2: List the six values in order.
Arranging from smallest to largest: \(B = 12 < D = 14 < A = 17 < E = 19 < C = 23 < F = 28\). So the smallest value is \(B = 12\) and the largest is \(F = 28\).

Step 3: Check option (AA), "D is the lowest integer and D = 14".
\(D = 14\) is correct, but \(D\) is not the lowest value; \(B = 12\) is lower. So option (AA) is false.

Step 4: Check option (BB), "C is the greatest integer and C = 23".
\(C = 23\) is correct, but \(C\) is not the greatest; \(F = 28\) is greater. So option (BB) is false.

Step 5: Check option (CC), "B is the lowest integer and B = 12".
Both parts hold: \(B = 12\) is indeed the smallest of all six values. So option (CC) is true.

Step 6: Check option (DD), "F is the greatest integer and F = 24".
\(F\) is indeed the greatest value, but its actual value is \(28\), not \(24\), so option (DD) is false.

Final Answer:
Only option (CC) has both parts correct.
\[ \boxed{\text{B is the lowest integer and } B = 12} \]