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12 Jan 2026, 11:51
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What is the greatest possible common divisor of two different positive integers, each less than 213, if neither of them is even?
A. 69
B. 71
C. 105
D. 207
E. 209
A. 69
B. 71
C. 105
D. 207
E. 209
12 Jan 2026, 11:51
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Bookmarks
Official Solution:
What is the greatest possible common divisor of two different positive integers, each less than 213, if neither of them is even?
A. 69
B. 71
C. 105
D. 207
E. 209
We want to find the greatest possible common divisor, say \(d\), of two different odd positive integers, both less than 213.
To maximize \(d\), we make the smaller number \(d\) itself and the larger number \(3d\) (since \(2d\) would be even, we skip it). This gives us the closest pair, allowing us to maximize \(d\). Thus, we need such \(3d\) that it is less than 213 and odd.
The greatest odd multiple of 3 less than 213 is 207, so \(3d = 207\), and \(d = \frac{207}{3} = 69\).
Answer: A
What is the greatest possible common divisor of two different positive integers, each less than 213, if neither of them is even?
A. 69
B. 71
C. 105
D. 207
E. 209
We want to find the greatest possible common divisor, say \(d\), of two different odd positive integers, both less than 213.
To maximize \(d\), we make the smaller number \(d\) itself and the larger number \(3d\) (since \(2d\) would be even, we skip it). This gives us the closest pair, allowing us to maximize \(d\). Thus, we need such \(3d\) that it is less than 213 and odd.
The greatest odd multiple of 3 less than 213 is 207, so \(3d = 207\), and \(d = \frac{207}{3} = 69\).
Answer: A
