Question

The addition of 7 distinct positive integers is 1740. What is the largest possible “greatest common divisor” of these 7 distinct positive integers?

• A. 42
• B. 60
• C. 74
• D. 140
• E. None of the above.

Answer 1

The Correct Option is B

To solve this problem, we need to find the greatest common divisor (GCD) of the 7 distinct positive integers whose sum is 1740. Our objective is to maximize this GCD. 

1. Let these integers be \(a_1, a_2, a_3, a_4, a_5, a_6, a_7\), and let their GCD be \(g\). Thus, each \(a_i\) can be expressed as \(a_i = g \times b_i\), where \(b_i\) are distinct positive integers.

2. Substituting the expressions into the sum yields: \(g \times b_1 + g \times b_2 + g \times b_3 + g \times b_4 + g \times b_5 + g \times b_6 + g \times b_7 = 1740\).

3. Factoring out \(g\) gives: \(g \times (b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7) = 1740\), which implies \(g \times S = 1740\) where \(S = b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7\).

4. To maximize \(g\), \(S\) should be minimized. Since \(b_i\) are distinct positive integers, the minimum value of \(S\) with 7 terms is achieved with \(b_i = 1, 2, 3, 4, 5, 6, 7\), giving \(S = 28\).

5. If \(S = 28\), then \(g = \frac{1740}{28} = 60\).

6. Therefore, with this setup, the largest possible GCD of these integers is 60.

The correct answer is 60.