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

Video Explanation

Explanatory Answer

Let the GCD of the 7 different positive numbers be x.
Then the sum of these 7 integers is at least 28x.
Think Why!?
The positive integers could be x, 2x, 3x, 4x, 5x, 6x, 7x
In this case The sum of the 7 integers is \( x \times \frac { 7 ( 7 + 1 ) } { 2 } = 28 x \)
In any other case, the sum will be some \( N \times y \), where y is the GCD and N is greater than 28.
Realize that the sum is fixed,
\( 1740 = 28 x = N \times y \)
So, the maximum value of the GCD occurs when N is as close to 28 as possible.
(1740 is not a multiple of 28.)
29 is a multiple of 1740.
When N = 29, y = 60
So, the largest GCD of these 7 distinct positive integers is 60.

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

Hence, the answer is '60'

Choice B is the correct answer.

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